Q. 38 Calculate areas and average valu... [FREE SOLUTION]

Chapter 4: Q. 38 (page 405)

Calculate areas and average values : Express each of the following areas or average values with a definite integral, and then use definite integral formulas to compute the exact value of the area or average value.
The area between the graphs of $f (x) = 4 - x^{2} .$ and $1 - 2 x$ on $[- 4, 4] .$

Short Answer

Expert verified

The area between $f (x)$ and $1 - 2 x$ is $24 .$

Step by step solution

Step 1. Given information

The given function is $f (x) = 4 - x^{2},$ we need to determine the area between the graph of $f (x)$ and $1 - 2 x$ on $[- 4, 4] .$

Step 2. Calculation

The area between $f (x)$ and $1 - 2 x$ on $[- 4, 4]$ is,

$|{鈭�}_{- 4}^{4} [(4 - x^{2}) - (1 - 2 x)] d x| .$

$= {鈭�}_{- 4}^{0} (3 - x^{2} + 2 x) d x + {鈭�}_{0}^{4} (3 - x^{2} + 2 x) d x .$

$= {[3 x - \frac{x^{3}}{3} + x^{2}]}_{- 4}^{0} + {[3 x - \frac{x^{3}}{3} + x^{2}]}_{0}^{4} .$

$= [0 - 0 + 0 - 3 (- 4) + \frac{{(4)}^{3}}{3} + {(- 4)}^{2}] + [3 (4) - \frac{{(4)}^{3}}{3} + {(4)}^{2}] .$

$= 12 + \frac{64}{3} - 16 + 12 - \frac{64}{3} + 16 .$

$= 24 .$

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91影视

Calculate areas and average values : Express each of the following areas or average values with a definite integral, and then use definite integral formulas to compute the exact value of the area or average value.
The area between the graphs of $f (x) = 4 - x^{2} .$ and $1 - 2 x$ on $[- 4, 4] .$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

One App. One Place for Learning.

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91影视

Calculate areas and average values : Express each of the following areas or average values with a definite integral, and then use definite integral formulas to compute the exact value of the area or average value. The area between the graphs of fx=4-x2.and 1-2xon -4,4.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Calculus

Mechanics Maths

Decision Maths

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

Calculate areas and average values : Express each of the following areas or average values with a definite integral, and then use definite integral formulas to compute the exact value of the area or average value.
The area between the graphs of $f (x) = 4 - x^{2} .$ and $1 - 2 x$ on $[- 4, 4] .$