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Chapter 5: Q. 93 (page 466)

Prove that the area of a circle of radius $r is 蟺 r^{2}$ , as follows:
(a) Write down a definite integral that represents the area of the circle of radius $r$ centered at the origin. (Hint: The equation of such a circle is $x^{2} + y^{2} = r^{2}$ .)
(b) Use trigonometric substitution to solve the definite integral you found in part (a). (Hint: Change the limits of integration to match your substitution .)

Short Answer

Expert verified

Part (a) The integral is $2 {鈭�}_{- r}^{r} \sqrt{r^{2} - x^{2}} d x$ .

Part (b) The value is $蟺 r^{2} .$

Step by step solution

01

Part (a) Step 1. Given information.

The equation of the circle is $x^{2} + y^{2} = r^{2}$ .

02

Part (a) Step 2. Explanation.

The definite integral will be,

$x^{2} + y^{2} = r^{2} y^{2} = r^{2} - x^{2} y = \sqrt{r^{2} - x^{2}} 2 {鈭�}_{- r}^{r} \sqrt{r^{2} - x^{2}} d x$

03

Part (b) Step 1. Explanation.

On solving,

$2 {鈭�}_{- r}^{r} \sqrt{r^{2} - x^{2}} d x = 2 {鈭�}_{- \frac{蟺}{2}}^{\frac{蟺}{2}} \sqrt{r^{2} - (r^{2} \sin^{2} u)} (r \cos u) d u = 2 r^{2} {鈭�}_{- \frac{蟺}{2}}^{\frac{蟺}{2}} \cos^{2} u d u = 2 r^{2} (\frac{1}{2}) {鈭�}_{- \frac{蟺}{2}}^{\frac{蟺}{2}} (1 + \cos 2 u) d u = r^{2} {[u + \frac{1}{2} \sin (2 u)]}_{- \frac{蟺}{2}}^{\frac{蟺}{2}} = {蟺谤}^{2}$

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Most popular questions from this chapter

Solve given definite integral.

${鈭�}_{1 / 4}^{1 / 2} 鈥� \frac{1}{x^{2} \sqrt{1 鈭� x^{2}}} d x$

Which of the integrals that follow would be good candidates for trigonometric substitution? If a trigonometric substitution is a good strategy, name the substitution. If another method is a better strategy, explain that method.

$(a) 鈭� \frac{4 + x^{2}}{x} d x$ $(b) 鈭� \frac{x}{4 + x^{2}} d x$

role="math" localid="1648759296940" $(c) 鈭� \frac{x^{2}}{4 + x^{2}} d x$ $(d) 鈭� \frac{16 鈭� x^{4}}{4 + x^{2}} d x$

Why doesn鈥檛 the definite integral ${鈭�}_{2}^{3} \sqrt{1 - x^{2}} d x$ make sense? (Hint: Think about domains.)

Complete the square for each quadratic in Exercises 28鈥�33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

$x^{2} - 5 x + 1$

Solve the integral: $鈭� x^{2} e^{3 x} d x$ .

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