For the Riemann sum on an interval [a, b], determine n, b, and f (x). Suppose that the interval 0 ≤ x ≤ 1 is divided into 100 subintervals of width Δx = .01. Show that the following sum is close to 5/4. [2(.01) + (.01)3]Δx + [2(.02) + (.02)3] Δx + … + [2(1.0) + (1.0)3] Δx.
> Find the area of the shaded region. 1.21 3 = fi X} = f 0 "
> Find the area of the shaded region. y=e-z Y 0 1 y=e² In 2 1
> Find the area of the shaded region. -2 =R Y 0 y = x³ + 2x 2
> Find the area of the shaded region. x x=k xt = f 0 fi
> Find the area under the curve y = 1 + √x from x = 1 to x = 9.
> Find the area under the curve y = (3x - 2)-3 from x = 1 to x = 2.
> Calculate the following integrals. 0∫1 (3 + e2x)/ex dx
> Calculate the following integrals. 0∫ln 3 (ex + e-x)/ e2x dx
> Calculate the following integrals. ln 2∫ln 3 (ex + e-x) dx
> Find the value of k that makes the antidifferentiation formula true. ∫ (5x - 7)-2 dx = k(5x - 7)-1 + C
> Calculate the following integrals. 0∫ln 2 (ex - e-x) dx
> Calculate the following integrals. -2∫2 3/2e3x dx
> Calculate the following integrals. 0∫5 (5 + 3x)-1 dx
> Calculate the following integrals. 3∫6 e2-(x/3) dx
> Calculate the following integrals. 1∫4 1/x2 dx
> Calculate the following integrals. 2/3 0∫8 √ (x + 1) dx
> Calculate the following integrals. 1∫24/x5 dx
> Calculate the following integrals. 0∫1 [2/(x + 1) – 1/(x + 4)] dx
> Calculate the following integrals. -1∫2 √ (2x + 4) dx
> Calculate the following integrals. 0∫1/8 5√x dx
> Find the value of k that makes the antidifferentiation formula true. ∫4 e3x+1 dx = k e3x+1 + C
> Calculate the following integrals. -1∫1 (x + 1)2 dx
> Calculate the following integrals. ∫ (5/x – x/5) dx
> Calculate the following integrals. ∫ √(4 – x) dx
> Calculate the following integrals. ∫ (2x + 3)7 dx
> Calculate the following integrals. ∫ (3x4 - 4x3) dx
> Calculate the following integrals. ∫5/√(x – 7) dx
> Calculate the following integrals. ∫ e-x/2 dx
> Calculate the following integrals. ∫ 5√(x + 3) dx
> What is a Riemann sum?
> In the formula Δx = (b – a)/n, what do a, b, n, and Δx denote?
> Find the value of k that makes the antidifferentiation formula true. ∫2e4x-1 dx = ke4x-1 + C
> State the formula for 1h(x)dx for each of the following functions. (a) h(x)xr, r ≠ -1 (b) h(x) = ekx (c) h(x) = 1/x (d) h(x) = f (x) + g(x) (e) h(x) = kf (x)
> What does it mean to anti-differentiate a function?
> State the formula for each of the following quantities: (a) average value of a function (b) consumers’ surplus (c) future value of an income stream (d) volume of a solid of revolution
> Outline a procedure for finding the area of a region bounded by two curves.
> How is F (x) |ba calculated, and what is it called?
> State the fundamental theorem of calculus.
> What is the difference between a definite integral and an indefinite integral?
> What is a definite integral?
> Give an interpretation of the area under a rate of change function. Give a concrete example.
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 100e-0.5x; a = 0, b = 4.
> Find the value of k that makes the antidifferentiation formula true. ∫3et/10 dt = ket/10 + C
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 1 - x; a = -1, b = 1.
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = x2; a = 0, b = 3.
> The following exercises ask for an unknown quantity x. After setting up the appropriate formula involving a definite integral, use the fundamental theorem to evaluate the definite integral as an expression in x. Because the resulting equation will be too
> The following exercises ask for an unknown quantity x. After setting up the appropriate formula involving a definite integral, use the fundamental theorem to evaluate the definite integral as an expression in x. Because the resulting equation will be too
> The following exercises ask for an unknown quantity x. After setting up the appropriate formula involving a definite integral, use the fundamental theorem to evaluate the definite integral as an expression in x. Because the resulting equation will be too
> The following exercises ask for an unknown quantity x. After setting up the appropriate formula involving a definite integral, use the fundamental theorem to evaluate the definite integral as an expression in x. Because the resulting equation will be too
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). Suppose that the interval 0 ≤ x ≤ 3 is divided into 100 subintervals of width Δx = .03. Let x1, x2, … , x100 be points in these subintervals. Suppose that in a particular application w
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). [3(.3)2 + 3(.9)2 + 3(1.5)2 + 3(2.1)2 + 3(2.7)2] (.6); a = 0
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). [(5 + e5) + (6 + e6) + (7 + e7)] (1); a = 4
> Find the value of k that makes the antidifferentiation formula true. ∫5e-2t dt = ke-2t + C
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). [3/1 + 3/1.5 + 3/2 + 3/2.5 + 3/3 + 3/3.5] (.5); a = 1
> For the Riemann sum on an interval [a, b], determine n, b, and f (x). [(8.25)3 + (8.75)3 + (9.25)3 + (9.75)3](.5); a = 8
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = e-x from x = 0 to x = 1
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = 2x + 1 from x = 0 to x = 1 (The solid generated is called a truncated cone.)
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = 2x - x2 from x = 0 to x = 2
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = √x from x = 0 to x = 4 (The solid generated is called a paraboloid.)
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = kx from x = 0 to x = h (generates a cone)
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = x2 from x = 1 to x = 2
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = √ (r2 - x2) from x = -r to x = r (generates a sphere of radius r)
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = √ (4 - x2) from x = -2 to x = 2 (generates a sphere of radius 2)
> Determine the following: ∫ (-3e-x + 2x - e0.5x/2) dx
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = -x2 + 1 from x = 0 to x = 1.
> Find the volume of the solid of revolution generated by revolving about the x-axis the region under each of the following curves. y = x + 1 from x = 0 to x = 2.
> A savings account pays 4.25% interest compounded continuously. At what rate per year must money be deposited steadily in the account to accumulate a balance of $100,000 after 10 years?
> An investment pays 10% interest compounded continuously. If money is invested steadily so that $5000 is deposited each year, how much time is required until the value of the investment reaches $140,000?
> Suppose that money is deposited steadily in a savings account so that $14,000 is deposited each year. Determine the balance at the end of 6 years if the account pays 4.5% interest compounded continuously.
> Suppose that money is deposited steadily in a savings account so that $16,000 is deposited each year. Determine the balance at the end of 4 years if the account pays 8% interest compounded continuously.
> Suppose that money is deposited daily in a savings account at an annual rate of $2000. If the account pays 6% interest compounded continuously, approximately how much will be in the account at the end of 2 years?
> Suppose that money is deposited daily in a savings account at an annual rate of $1000. If the account pays 5% interest compounded continuously, estimate the balance in the account at the end of 3 years.
> For a particular commodity, the quantity produced and the unit price are given by the coordinates of the point where the supply and demand curves intersect. For the pair of supply and demand curves, determine the point of intersection (A, B) and the cons
> For a particular commodity, the quantity produced and the unit price are given by the coordinates of the point where the supply and demand curves intersect. For the pair of supply and demand curves, determine the point of intersection (A, B) and the cons
> Determine the following: ∫-2(e2x + 1) dx
> Figure 8 shows a supply curve for a commodity. It gives the relationship between the selling price of the commodity and the quantity that producers will manufacture. At a higher selling price, a greater quantity will be produced. Therefore, the curve is
> Figure 8 shows a supply curve for a commodity. It gives the relationship between the selling price of the commodity and the quantity that producers will manufacture. At a higher selling price, a greater quantity will be produced. Therefore, the curve is
> Figure 8 shows a supply curve for a commodity. It gives the relationship between the selling price of the commodity and the quantity that producers will manufacture. At a higher selling price, a greater quantity will be produced. Therefore, the curve is
> Figure 8 shows a supply curve for a commodity. It gives the relationship between the selling price of the commodity and the quantity that producers will manufacture. At a higher selling price, a greater quantity will be produced. Therefore, the curve is
> Find the consumers’ surplus for each of the following demand curves at the given sales level, x. p = √(16 - .02x); x = 350
> Find the consumers’ surplus for each of the following demand curves at the given sales level, x. p = 500/(x + 10) - 3; x = 40
> Find the consumers’ surplus for each of the following demand curves at the given sales level, x. p = x2/200 - x + 50; x = 20
> Find the consumers’ surplus for each of the following demand curves at the given sales level, x. p = 3 – x/10; x = 20
> One hundred dollars is deposited in the bank at 5% interest compounded continuously. What will be the average value of the money in the account during the next 20 years?
> One hundred grams of radioactive radium having a half-life of 1690 years is placed in a concrete vault. What will be the average amount of radium in the vault during the next 1000 years?
> Determine the following: ∫7/(2e2x) dx
> Assuming that a country’s population is now 3 million and is growing exponentially with growth constant .02, what will be the average population during the next 50 years?
> During a certain 12-hour period, the temperature at time t (measured in hours from the start of the period) was T(t) = 47 + 4t – 1/3 t2 degrees. What was the average temperature during that period?
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 1/√x; a = 1, b = 9.
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 1/x; a = 1/3, b = 3.
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 2; a = 0, b = 1.
> Shade the portion of Fig. 23 whose area is given by the integral 0∫2 [ f (x) - g (x)] dx + 2∫4 [h(x) - g (x)] dx. Figure 23: 3 1 y = h(x) y = g(x) 2 3 y = f(x) 4 5
> Write a definite integral or sum of definite integrals that gives the area of the shaded portions in Fig. 22. Figure 22: 2 1 U 0 -1 -2- 1 y = f(x) 不 3 4 y = g(r) M
> Write a definite integral or sum of definite integrals that gives the area of the shaded portions in Fig. 21. Figure 21: -1 0 y y = f(x) 2 3 4
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = 1/x, y = 3 - x
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = √(x + 1), y = (x - 1)2
> Determine the following: ∫e dx
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = 5 - (x - 2)2, y = ex
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = ex, y = 4x + 1
> The velocity of an object moving along a line is given by υ(t) = t2 + t - 2 feet per second. (a) Find the displacement of the object as t varies in the interval 0 ≤ t ≤ 3. Interpret this displacement using area under the graph of υ(t). (b) Find the total
