### Problem Statement2. Compute the value of the integral $\int_{0}^{5} \sqrt{25-x^2} \, dx$.### ExplanationThis integral represents the area under the curve of the function $y = \sqrt{25-x^2}$ from $x = 0$ to $x = 5$.### Geometric InterpretationThe function $y = \sqrt{25-x^2}$ is the upper half of a circle with a radius of 5 centered at the origin in the Cartesian plane. The integral calculates the area of the quarter-circle in the first quadrant of this circle.

Given data: The given integral expression is I=∫0525-x2dx. Assume x=5sint, and differentiate…

Answered: 2. Compute the value of | V25–x² dr

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...

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### Problem Statement

2. Compute the value of the integral $\int_{0}^{5} \sqrt{25-x^2} \, dx$.

### Explanation

This integral represents the area under the curve of the function $y = \sqrt{25-x^2}$ from $x = 0$ to $x = 5$.

### Geometric Interpretation

The function $y = \sqrt{25-x^2}$ is the upper half of a circle with a radius of 5 centered at the origin in the Cartesian plane. The integral calculates the area of the quarter-circle in the first quadrant of this circle.

Expert Solution

Step 1

Given data:

The given integral expression is $I = \int_{0}^{5} \sqrt{25 - x^{2}} d x$ .

Assume x=5sint, and differentiate it with respect to x.

$\begin{array}{rcl} \frac{d x}{d x} & = & \frac{d}{d x} (5 \sin t) \\ d x & = & (5 \cos t) d t \end{array}$

Evaluate the lower and upper limits of t.

$\begin{array}{rcl} 0 & = & 5 \sin t_{1} \\ \sin t_{1} & = & 0 \\ t_{1} & = & 0 \\ 5 & = & 5 \sin t_{2} \\ \sin t_{2} & = & 1 \\ t_{2} & = & \frac{π}{2} \end{array}$

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