Custom Solutions to Stewart Calculus, Integral
Answer
Sure, you came to the right place to get custom solutions given by experts.
For this integral we start by integrating with respect to $z$ : $$ \int _0 ^1 \int _0 ^1 \int_0 ^{2-x^2-y^2} xy e^{z}\,dz\,dy\,dx = \int _0 ^1 \int _0 ^1 xy e^{z}\,\Big]_0 ^{2-x^2-y^2}\,dy\,dx \\ = \int _0 ^1 \int _0 ^1 xy( e^{2-x^2-y^2}-1)\,dy\,dx. $$
Then we do the $y$ integral. For this let $u=2-x^2-y^2$ to get $du=-2ydy$ (remember that in the $y$ integral we treat $x$ as a constant!)
$$ \int _0 ^1 \int _0 ^1 xy( e^{2-x^2-y^2}-1)\,dy\,dx \\ \; \\ =\int _0 ^1 \int _0 ^1 \frac{-1}{2}x (e^{u}-1)\,du\,dx = \int _0 ^1 \frac{-1}{2}x (e^{u}-u)\,\Big]_0 ^1\,du\,dx \\ \; \\ =\int _0 ^1 \frac{-1}{2}x (e-2)\,dx = \frac{-(e-2)}{2}\,\frac{x^2}{2}\,\Big]_0 ^1 = \frac{-(e-2)}{4}$$
- accepted
- 425 views
- $15.00
Related Questions
- Use Green’s theorem to compute $\int_C x^2 ydx − xy^2 dy$ where $C$ is the circle $x^2 + y ^2 = 4$ oriented counter-clockwise.
- Does $\lim_{(x,y)\rightarrow (0,0)}\frac{(x^2-y^2) \cos (x+y)}{x^2+y^2}$ exists?
- Proving f is continuous
- Solutions to Stewart Calculus 8th edition
- Evaluate $\int ...\int_{R_n}dV_n(x_1^2 + x_2^2 + ... + x_n^2)$ , where $n$ and $R_n$ is defined in the body of this question.
- Prove the trig identity $\sec x- \sin x \tan x =\frac{1}{\sec x}$
- The volume of a spherical tank with radius = r is expanding in such a way that r is increasing at 1 cm/min. At what rate is the volume expanding when r = 100 cm?
- Fermat's method of calculus