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Chapter 16: Problem 12

Evaluate the following integrals. A sketch of the region of integration may be useful. $$\int_{0}^{2} \int_{1}^{2} \int_{0}^{1} y z e^{x} d x d z d y$$

Short Answer

Expert verified
Question: Evaluate the triple integral $$\int_{0}^{2} \int_{1}^{2} \int_{0}^{1} y z e^{x} d x d z d y$$. Answer: The value of the triple integral is \(3(e-1)\).

Step by step solution

01

Write down the given integral

We are given the following triple integral: $$\int_{0}^{2} \int_{1}^{2} \int_{0}^{1} y z e^{x} d x d z d y$$
02

Perform the first integral with respect to x

We will evaluate the triple integral step by step starting with the x integral: $$\int_{0}^{2} \int_{1}^{2} \left[ \int_{0}^{1} y z e^{x} d x \right] d z d y$$ Now, perform the x-integral: $$\int_{0}^{2} \int_{1}^{2} yze^{x}\Big|_0^1 d z d y$$
03

Evaluate the x boundaries

Evaluate the expression with the x boundaries. $$\int_{0}^{2} \int_{1}^{2} yz(e^1 - e^0) d z d y$$ Since \(e^0=1\), we can further simplify: $$\int_{0}^{2} \int_{1}^{2} yz(e - 1) d z d y$$
04

Perform the second integral with respect to z

Now, let's focus on the z-integral: $$\int_{0}^{2} \left[ \int_{1}^{2} yz(e - 1) d z\right] d y$$ Perform the z-integral: $$\int_{0}^{2} \frac{1}{2}y(e-1)z^2\Big|_1^2 d y$$
05

Evaluate the z boundaries

Evaluate the expression with the z boundaries. $$\int_{0}^{2} \frac{1}{2}y(e-1)(4-1) d y$$ Further simplify: $$\int_{0}^{2} \frac{3}{2}y(e-1) d y$$
06

Perform the third integral with respect to y

Finally, let's perform the y-integral: $$\left[ \frac{3}{4}(e-1)y^2 \right]_0^2$$
07

Evaluate the y boundaries

Evaluate the expression with the y boundaries. $$\frac{3}{4}(e-1)(4-0)$$
08

Simplify the result

Simplify the expression to find the final answer. $$3(e-1)$$ So, the value of the triple integral is \(3(e-1)\).

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