
# 7.4E:Exercises


## Exercise $$\PageIndex{1}$$

In the following exercises, evaluate the triple integrals over the rectangular solid box $$B$$.

1. $\iiint_B (2x + 3y^2 + 4z^3) \space dV,$ where $$B = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq 2, \space 0 \leq z \leq 3\}$$

$$192$$

2. $\iiint_B (xy + yz + xz) \space dV,$ where $$B = \{(x,y,z) | 1 \leq x \leq 2, \space 0 \leq y \leq 2, \space 1 \leq z \leq 3\}$$

3. $\iiint_B (x \space cos \space y + z) \space dV,$ where $$B = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq \pi, \space -1 \leq z \leq 1\}$$

$$0$$

4. $\iiint_B (z \space sin \space x + y^2) \space dV,$ where $$B = \{(x,y,z) | 0 \leq x \leq \pi, \space 0 \leq y \leq 1, \space -1 \leq z \leq 2\}$$

## Exercise $$\PageIndex{2}$$

In the following exercises, change the order of integration by integrating first with respect to $$z$$, then $$x$$, then $$y$$.

1. $\int_0^1 \int_1^2 \int_2^3 (x^2 + ln \space y + z) \space dx \space dy \space dz$

$\int_0^1 \int_1^2 \int_2^3 (x^2 + ln \space y + z) \space dx \space dy \space dz = \frac{35}{6} + 2 \space ln 2$

2. $\int_0^1 \int_{-1}^1 \int_0^3 (ze^x + 2y) \space dx \space dy \space dz$

3. $\int_{-1}^2 \int_1^3 \int_0^4 \left(x^2z + \frac{1}{y}\right) \space dx \space dy \space dz$

$\int_{-1}^2 \int_1^3 \int_0^4 \left(x^2z + \frac{1}{y}\right) \space dx \space dy \space dz = 64 + 12 \space ln \space 3$

4. $\int_1^2 \int_{-2}^{-1} \int_0^1 \frac{x + y}{z} \space dx \space dy \space dz$

## Exercise $$\PageIndex{3}$$

1. Let $$F$$, $$G$$, and $$H$$ be continuous functions on $$[a,b]$$, $$[c,d]$$, and $$[e,f]$$, respectively, where $$a, \space b, \space c, \space d, \space e$$, and $$f$$ are real numbers such that $$a < b, \space c < d$$, and $$e < f$$. Show that

$\int_a^b \int_c^d \int_e^f F (x) \space G (y) \space H(z) \space dz \space dy \space dx = \left(\int_a^b F(x) \space dx \right) \left(\int_c^d G(y) \space dy \right) \left(\int_e^f H(z) \space dz \right).$

2. Let $$F$$, $$G$$, and $$H$$ be differential functions on $$[a,b]$$, $$[c,d]$$, and $$[e,f]$$, respectively, where $$a, \space b, \space c, \space d, \space e$$, and $$f$$ are real numbers such that $$a < b, \space c < d$$, and $$e < f$$. Show that

$\int_a^b \int_c^d \int_e^f F' (x) \space G' (y) \space H'(z) \space dz \space dy \space dx = [F (b) - F (a)] \space [G(d) - G(c)] \space H(f) - H(e)].$

## Exercise $$\PageIndex{4}$$

In the following exercises, evaluate the triple integrals over the bounded region $$E = \{(x,y,z) | a \leq x \leq b, \space h_1 (x) \leq y \leq h_2 (x), \space e \leq z \leq f \}.$$

1. $\iiint_E (2x + 5y + 7z) \space dV,$ where $$E = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq -x + 1, \space 1 \leq z \leq 2\}$$

$$\frac{77}{12}$$

2. $\iiint_E (y \space ln \space x + z) \space dV,$ where $$E = \{(x,y,z) | 1 \leq x \leq e, \space 0 \leq y ln \space x, \space 0 \leq z \leq 1\}$$

$\iiint_E (sin \space x + sin \space y) dV,$ where $$E = \{(x,y,z) | 0 \leq x \leq \frac{\pi}{2}, \space -cos \space x \leq y cos \space x, \space -1 \leq z \leq 1 \}$$

$$2$$

3. $\iiint_E (xy + yz + xz ) dV$ where $$E = \{(x,y,z) | 0 \leq x \leq 1, \space -x^2 \leq y \leq x^2, \space 0 \leq z \leq 1 \}$$

In the following exercises, evaluate the triple integrals over the indicated bounded region $$E$$.

4. $\iiint_E (x + 2yz) \space dV,$ where $$E = \{(x,y,z) | 0 \leq x \leq 1, \space 0 \leq y \leq x, \space 0 \leq z \leq 5 - x - y \}$$

$$\frac{430}{120}$$

5. $\iiint_E (x^3 + y^3 + z^3) \space dV,$ where $$E = \{(x,y,z) | 0 \leq x \leq 2, \space 0 \leq y \leq 2x, \space 0 \leq z \leq 4 - x - y \}$$

6. $\iiint_E y \space dV,$ where $$E = \{(x,y,z) | -1 \leq x \leq 1, \space -\sqrt{1 - x^2} \leq y \leq \sqrt{1 - x^2}, \space 0 \leq z \leq 1 - x^2 - y^2 \}$$

$$0$$

7. $\iiint_E x \space dV,$ where $$E = \{(x,y,z) | -2 \leq x \leq 2, \space -4\sqrt{1 - x^2} \leq y \leq \sqrt{4 - x^2}, \space 0 \leq z \leq 4 - x^2 - y^2 \}$$

## Exercise $$\PageIndex{5}$$

In the following exercises, evaluate the triple integrals over the bounded region $$E$$ of the form $$E = \{(x,y,z) | g_1 (y) \leq x \leq g_2(y), \space c \leq y \leq d, \space e \leq z \leq f \}$$.

1. $\iiint_E x^2 \space dV,$ where $$E = \{(x,y,z) | 1 - y^2 \leq x \leq y^2 - 1, \space -1 \leq y \leq 1, \space 1 \leq z \leq 2 \}$$

$$-\frac{64}{105}$$

2. $\iiint_E (sin \space x + y) \space dV,$ where $$E = \{(x,y,z) | -y^4 \leq x \leq y^4, \space 0 \leq y \leq 2, \space 0 \leq z \leq 4\}$$

3. $\iiint_E (x - yz) \space dV,$ where $$E = \{(x,y,z) | -y^6 \leq x \leq \sqrt{y}, \space 0 \leq y \leq 1x, \space -1 \leq z \leq 1 \}$$

$$\frac{11}{26}$$

4. $\iiint_E z \space dV,$ where $$E = \{(x,y,z) | 2 - 2y \leq x \leq 2 + \sqrt{y}, \space 0 \leq y \leq 1x, \space 2 \leq z \leq 3 \}$$

## Exercise $$\PageIndex{6}$$

In the following exercises, evaluate the triple integrals over the bounded region $$E = \{(x,y,z) | g_1(y) \leq x \leq g_2(y), \space c \leq y \leq d, \space u_1(x,y) \leq z \leq u_2 (x,y) \}$$

1. $\iiint_E z \space dV,$ where $$E = \{(x,y,z) | -y \leq x \leq y, \space 0 \leq y \leq 1, \space 0 \leq z \leq 1 - x^4 - y^4 \}$$

$$\frac{113}{450}$$

2. $\iiint_E (xz + 1) \space dV,$ where $$E = \{(x,y,z) | 0 \leq x \leq \sqrt{y}, \space 0 \leq y \leq 2, \space 0 \leq z \leq 1 - x^2 - y^2 \}$$

3. $\iiint_E (x - z) \space dV,$ where $$E = \{(x,y,z) | - \sqrt{1 - y^2} \leq x \leq y, \space 0 \leq y \leq \frac{1}{2}x, \space 0 \leq z \leq 1 - x^2 - y^2 \}$$

$$\frac{1}{160}(6 \sqrt{3} - 41)$$

4. $\iiint_E (x + y) \space dV,$ where $$E = \{(x,y,z) | 0 \leq x \leq \sqrt{1 - y^2}, \space 0 \leq y \leq 1x, \space 0 \leq z \leq 1 - x \}$$

In the following exercises, evaluate the triple integrals over the bounded region $$E = \{(x,y,z) | (x,y) \in D, \space u_1 (x,y) x \leq z \leq u_2 (x,y) \}$$, where $$D$$ is the projection of $$E$$ onto the $$xy$$-plane

5. $\iint_D \left(\int_1^2 (x + y) \space dz \right) \space dA,$ where $$D = \{(x,y) | x^2 + y^2 \leq 1\}$$

$$\frac{3\pi}{2}$$

6. $\iint_D \left(\int_1^3 x (z + 1)\space dz \right) \space dA,$ where $$D = \{(x,y) | x^2 -y^2 \geq 1, \space x \leq \sqrt{5}\}$$

7. $\iint_D \left(\int_0^{10-x-y} (x + 2z) \space dz \right) \space dA,$ where $$D = \{(x,y) | y \geq 0, \space x \geq 0, \space x + y \leq 10\}$$

$$1250$$
8. $\iint_D \left(\int_0^{4x^2+4y^2} y \space dz \right) \space dA,$ where $$D = \{(x,y) | x^2 + y^2 \leq 4, \space y \geq 1, \space x \geq 0\}$$
## Exercise $$\PageIndex{7}$$
1. The solid $$E$$ bounded by $$y^2 + z^2 = 9, \space z = 0$$, and $$x = 5$$ is shown in the following figure. Evaluate the integral