Use the integration capabilities of the graphing utility to approximate the volume of the solid generated by revolving the region about the y-axis.

$$y=\sqrt{1-x^3}, \quad y=0, \quad x=0$$

Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the y-axis. $y=\frac{x^{2}}{2}+4$, $0 \leq x \leq 2$

Use a graphing utility to graph the region bounded by the graphs of the functions.

$$f(x)=x^4-9 x^2, g(x)=x^3-9 x$$

The region bounded by the graphs of y = 2√x, y=0, x=3, and x=8 is revolved about the x-axis. Find the surface area of the solid generated.

(a) use a graphing utility to graph the region bounded by the graphs of the equations, (b) find the area of the region analytically, and (c) use the integration capabilities of the graphing utility to verify your results. f(x) = x^4 - 9x², g(x) = x³-9x

Use a graphing utility to graph the region bounded by the graphs of the equations,

$f(x)=x^4-9 x^2, \quad g(x)=x^3-9 x$

Let f be rectifiable on the interval [a, b], and let s(x) = ∫_a^x √1 + [f′(t)]² dt. Use the function f(t) = t^3/2 and interval [1, 3] to calculate s(2) and describe what it signifies.

Let f be rectifiable on the interval [a, b], and let s(x) = ∫_a^x √1+[f′(t)]² dt. (c) Find s(x) on [1, 3] when f(t) = t^3/2.

The Gateway Arch in St. Louis, Missouri, is modeled by y = 693.8597 - 68.7672 cosh 0.0100333x, -299.2239 ≤ x ≤ 299.2239. Use the integration capabilities of a graphing utility to approximate the length of this curve.

Let f be rectifiable on the interval [a, b], and let s(x) = ∫_a^x √1+[f′(t)]² dt. (b) Find ds and (ds)².

Which of the following requires more work? Explain your reasoning. (a) A 60-pound box of books is lifted 3 feet. (b) A 60-pound box of books is held 3 feet in the air for 2 minutes.

Set up the definite integral that gives the area of the region. y_1 = x² - 4x + 3, y_2 = -x² + 2x + 3

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given axis.

$y=\sqrt{6 x-6}, \quad y=\frac{1}{6}(x-1)^2, \quad x$-axis

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y = sin x, y = 0, x = 0, x = π