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Chapter 4: Q. 60 (page 373)

Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises $19 鈥� 64$ . Use
a graph to check your answer.
${鈭�}_{- 1}^{3} |4 - x^{2}| d x$

Short Answer

Expert verified

The value of integral is $\frac{34}{3}$ and the plot is

Step by step solution

01

Step 1. Given information

An expression is given as ${鈭�}_{- 1}^{3} |4 - x^{2}| d x$

02

Step 2. Evaluating integral

Simplify the expression as,

${鈭�}_{- 1}^{3} |4 - x^{2}| d x = {鈭�}_{- 1}^{0} 4 - x^{2} d x + {鈭�}_{0}^{2} x^{2} - 4 d x + {鈭�}_{2}^{3} 4 - x^{2} d x = {鈭�}_{- 1}^{0} 4 d x - {鈭�}_{- 1}^{0} x^{2} d x + {鈭�}_{0}^{2} x^{2} d x - {鈭�}_{0}^{2} 4 d x - {鈭�}_{2}^{3} 4 d x + {鈭�}_{2}^{3} x^{2} d x = 4 [x]_{- 1}^{0} - {[\frac{x^{3}}{3}]}_{- 1}^{0} + {[\frac{x^{3}}{3}]}_{0}^{2} - 4 [x]_{0}^{2} - 4 [x]_{2}^{3} + {[\frac{x^{3}}{3}]}_{2}^{3} = 4 (2 + 1) - (\frac{8}{3} + \frac{1}{3}) - 4 (3 - 2) + (\frac{27}{3} - \frac{8}{3}) = 12 - \frac{9}{3} - 4 + \frac{19}{3} = 8 + \frac{10}{3} = \frac{24 + 10}{3} = \frac{34}{3}$

And the plot is

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Most popular questions from this chapter

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value

$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} {(1 + \frac{k}{n})}^{2} . \frac{1}{n}$

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