Write and evaluate the definite integral that represents the area of the surface generated by revolving the curve on the indicated interval about the x-axis. $y=\frac{x^{3}}{18}$, $3 \leq x \leq 6$

Use the sum of the first 10 terms to estimate the sum of the series summation n=1 to infinity 1/n^2.How good is this estimate?

(a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hopital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). $\lim _{x \rightarrow 0^{+}} x^{1 / x}$

(a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). lim_(x→0^+) [cos(π/2 - x)]^x

Find the area of the surface generated by revolving the curve about the indicated axis. x = y³/3, 0 ≤ y ≤ 1; y-axis

Determine the first four terms of the Maclaurin series for e^2x by using the definition of the Maclaurin series and the formula for the coefficient of the n term, a_n = f^(n) (0)/n!.

Determine all values of p for which the improper integral converges. ∫_1^∞ 1/x^p dx

Given the region bounded by the graphs of y = ln x, y = 0, and x = e find (a) the area of the region. (b) the volume of the solid generated by revolving the region about the x-axis. (c) the volume of the solid generated by revolving the region about the y-axis. (d) the centroid of the region.

Find the area of the region. Interior of r=3 cos θ

Find the equations of the tangent lines at the point where the curve crosses itself. x = 2 - π cos t, y = 2t - π sin t

Find dy/dx using logarithmic differentiation. y = x^(x − 7), x > 0

Find the time required for an object to cool from 300°F to 250°F by evaluating t = 10/ln 2 ∫_250^300 1/T-100 dT where t is time in minutes.

Find M_x, M_y, and (x̅, y̅) for the laminas of uniform density ρ bounded by the graphs of the equations. x = y + 2, x = y²

Decide whether it is more convenient to use the disk method or the shell method to find the volume of the solid of revolution. Explain your reasoning. y=4-e^x