Set up the integral that gives the volume of the solid. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice. (Type exact answers.) A. \( \int\left(2 \pi\left(x^{2}\right)^{2} d x\right. \) B. \( \int\left(x^{2}\right) \mathrm{dx} \) The volume of the solid is (Type an exact answer.)

(a) \( \sum_{n=1}^{\infty} 2\left(\frac{4}{3}\right)^{n-1} \)

Suppose \( \int_{2}^{4} f(x) d x=-8 \int_{2}^{7} f(x) d x=6 \), and \( \int_{2}^{7} g(x) d x=5 \) Evaluate the following integrals \( \int_{7}^{2} g(x) d x=\square \) \( \int_{2}^{7} 6 g(x) d x=\square \) (Simplify your answer.) \( \int_{2}^{7}[g(x)-f(x)] d x=\square \) (Simplify your answer )

\( \int _ { 0 } ^ { 3 \pi / 2 } x \sin x d x = \square \)

b) \( \lim _{x \rightarrow \frac{\pi}{4}} \frac{\sqrt{2} \sin x-1}{2 \cos ^{2} x-1} \)

Complete the following steps for the given function interval, and value of \( n \). a. Sketch the graph of the function on the given interval. b. Calculate \( \Delta x \) and the grid points \( x_{0} x_{1} \quad x_{n} \) c. Illustrate the left and right Riemann sums, and determine which Riemann sum underestimates and which sum overestimates the area under the curve d. Calculate the left and right Riemann sums. \[ f(x)=\cos (x-5 \pi / 8) \text { on }[\pi / 2 \pi] n=4 \]