Chapter 4: Q. 62 (page 373)

Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises $19 - 64$ . Use
a graph to check your answer.
$\int_{0}^{6} | (x - 1) (x - 4) | d x$

Short Answer

Expert verified

The value of integral is $15$ and the plot is

Step by step solution

Step 1. Given information

An expression is given as $\int_{0}^{6} | (x - 1) (x - 4) | d x$

Step 2. Evaluating integral

We have to change the integral as

$f (x) = \{\begin{array}{lc} (x - 1) (x - 4) & 0 \leq x \leq 1 \\ - (x - 1) (x - 4) & 1 \leq x \leq 4 \\ (x - 1) (x - 4) & 4 \leq x \leq 6 \end{array}$

Now evaluate the integral,

$\int_{0}^{6} | (x - 1) (x - 4) | d x = \int_{0}^{1} (x - 1) (x - 4) d x + \int_{1}^{4} - (x - 1) (x - 4) d x + \int_{4}^{6} (x - 1) (x - 4) d x = \int_{0}^{1} (x^{2} - 5 x + 4) d x - \int_{1}^{4} ((x^{2} - 5 x + 4)) d x + \int_{4}^{6} (x^{2} - 5 x + 4) d x = {(\frac{x^{3}}{3} - \frac{5}{2} x^{2} + 4 x)}_{0}^{1} - {(\frac{x^{3}}{3} - \frac{5}{2} x^{2} + 4 x)}_{1}^{4} + {(\frac{x^{3}}{3} - \frac{5}{2} x^{2} + 4 x)}_{4}^{6} = (\frac{1^{3}}{3} - \frac{5}{2} (1)^{2} + 4 (1) - 0) - ((\frac{4^{3}}{3} - \frac{5}{2} (4)^{2} + 4 (4)) - (\frac{1^{3}}{3} - \frac{5}{2} (1)^{2} + 4 (1))) + ((\frac{6^{3}}{3} - \frac{5}{2} (6)^{2} + 4 (6)) - (\frac{4^{3}}{3} - \frac{5}{2} (4)^{2} + 4 (4))) = \frac{11}{6} - (- \frac{9}{2}) + \frac{26}{3} = \frac{11}{6} + \frac{9}{2} + \frac{26}{3} = \frac{90}{6} = 15$

And the plot is

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Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises $19 - 64$ . Use
a graph to check your answer.
$\int_{0}^{6} | (x - 1) (x - 4) | d x$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating integral

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Use the Fundamental Theorem of Calculus to find the exactvalues of each of the definite integrals in Exercises 19–64. Usea graph to check your answer.∫06|(x-1)(x-4)|dx

Short Answer

Step by step solution

Step 1. Given information

Step 2. Evaluating integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Decision Maths

Probability and Statistics

Logic and Functions

Statistics

Study anywhere. Anytime. Across all devices.

Use the Fundamental Theorem of Calculus to find the exact
values of each of the definite integrals in Exercises $19 - 64$ . Use
a graph to check your answer.
$\int_{0}^{6} | (x - 1) (x - 4) | d x$