Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
5.1.1
![To solve the given double integral, we need to evaluate the integral:
\[
\int_{5}^{7} \int_{4}^{5} (x + y) \, dy \, dx
\]
Let's solve this step by step.
### Step 1: Integrate with respect to \( y \)
The inner integral is:
\[
\int_{4}^{5} (x + y) \, dy
\]
This can be split into two separate integrals:
\[
\int_{4}^{5} x \, dy + \int_{4}^{5} y \, dy
\]
The first integral evaluates to:
\[
x \cdot [y]_{4}^{5} = x(5 - 4) = x
\]
The second integral evaluates to:
\[
\left[ \frac{y^2}{2} \right]_{4}^{5} = \frac{5^2}{2} - \frac{4^2}{2} = \frac{25}{2} - \frac{16}{2} = \frac{9}{2}
\]
The result of the inner integral is:
\[
x + \frac{9}{2}
\]
### Step 2: Integrate with respect to \( x \)
Now evaluate the outer integral:
\[
\int_{5}^{7} \left(x + \frac{9}{2}\right) \, dx
\]
This can also be split into:
\[
\int_{5}^{7} x \, dx + \int_{5}^{7} \frac{9}{2} \, dx
\]
The first integral evaluates to:
\[
\left[ \frac{x^2}{2} \right]_{5}^{7} = \frac{7^2}{2} - \frac{5^2}{2} = \frac{49}{2} - \frac{25}{2} = \frac{24}{2} = 12
\]
The second integral evaluates to:
\[
\frac{9}{2} \cdot [x]_{5}^{7} = \frac{9}{2}(7 - 5) = \frac{9}{2} \cdot 2 = 9
\]
Adding these together gives:
\[
12 + 9 = 21
\]
Therefore, the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91ed8060-678c-4d36-a5c7-0e7a65a577a0%2Ffa214373-a05b-4c29-ad6d-ce5d159289ee%2F5h1kh6k_processed.png&w=3840&q=75)
Transcribed Image Text:To solve the given double integral, we need to evaluate the integral:
\[
\int_{5}^{7} \int_{4}^{5} (x + y) \, dy \, dx
\]
Let's solve this step by step.
### Step 1: Integrate with respect to \( y \)
The inner integral is:
\[
\int_{4}^{5} (x + y) \, dy
\]
This can be split into two separate integrals:
\[
\int_{4}^{5} x \, dy + \int_{4}^{5} y \, dy
\]
The first integral evaluates to:
\[
x \cdot [y]_{4}^{5} = x(5 - 4) = x
\]
The second integral evaluates to:
\[
\left[ \frac{y^2}{2} \right]_{4}^{5} = \frac{5^2}{2} - \frac{4^2}{2} = \frac{25}{2} - \frac{16}{2} = \frac{9}{2}
\]
The result of the inner integral is:
\[
x + \frac{9}{2}
\]
### Step 2: Integrate with respect to \( x \)
Now evaluate the outer integral:
\[
\int_{5}^{7} \left(x + \frac{9}{2}\right) \, dx
\]
This can also be split into:
\[
\int_{5}^{7} x \, dx + \int_{5}^{7} \frac{9}{2} \, dx
\]
The first integral evaluates to:
\[
\left[ \frac{x^2}{2} \right]_{5}^{7} = \frac{7^2}{2} - \frac{5^2}{2} = \frac{49}{2} - \frac{25}{2} = \frac{24}{2} = 12
\]
The second integral evaluates to:
\[
\frac{9}{2} \cdot [x]_{5}^{7} = \frac{9}{2}(7 - 5) = \frac{9}{2} \cdot 2 = 9
\]
Adding these together gives:
\[
12 + 9 = 21
\]
Therefore, the
Expert Solution
Step 1
Given:
To evaluate the given iterated integral.
Following formulas are used to evaluate the given integral:
Step by step
Solved in 3 steps
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