Answer to Question #276217 in Calculus for Ella

Question

1. Evaluate $\intop$x²(1 + 2x³)³dx.

2. Evaluate $\intop$xe^7x dx.

3. Find the volume of the solid of revolution when the curve y = 1 + x2 is revolved around the x-axis on [−2, 2].

Answer 1

(1) $I=\int x^{2}\left(1+2 x^{3}\right)^{3} d x$

By substitution method, let $u=1+2 x^{3}$

Such that $d u=6 x^{2} d x$

$\begin{aligned} &I=\int x^{2}\left(1+2 x^{3}\right)^{3} d x=I=\int x^{2} u^{3} \frac{d u}{6 x^{2}} \\ &I=\frac{1}{4} \int u^{3} d u=\frac{u^{4}}{24}+C \\ &I=\frac{\left(1+2 x^{3}\right)^{4}}{24}+C \end{aligned}$

(2) $I=\int x e^{7 x} d x$

Using integration by part

$\begin{aligned} &u=x \quad, \quad d v=e^{7 x} d x \\ &d u=1, \quad v=\frac{e^{7 x}}{7} \\ &\int u d v=u v-\int v d u \\ &I=\frac{x e^{7 x}}{7}-\frac{1}{7} \int e^{7 x} d x \end{aligned}$

$I=\frac{x e^{7 x}}{7}-\frac{e^{7 x}}{49}+C$

(3) $y=1+x^{2}$

Using method of disk

$\begin{aligned} &V=\int_{-2}^{2} \pi y^{2} d x=\int_{-2}^{2} \pi\left(1+x^{2}\right)^{2} d x \\ &V=\int_{-2}^{2} \pi\left(1+2 x^{2}+x^{4}\right) d x \\ &V=\frac{412 \pi}{15} \end{aligned}$