Answered: Please do #33. (1)First need to draw a…

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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Please do #33. (1)First need to draw a graph of the region — and copy it on you paper. (2) Show the algebra steps as need when (i) finding the solutions to f(x) = g(x) to get the intersection points of the two curves forming the region; and (ii) determining for EACH subregion that, which curve is the top boundary and which one is the bottom (that is checking if f(x) is greater or g(x) is greater). (3) Set up integral(s) to calculate the area of each region.

E Finding the Area of a Region In Exercises 31–36, (a) use
a graphing utility to graph the region bounded by the graphs
of the equations, (b) find the area of the region analytically, and
(c) use the integration capabilities of the graphing utility to
verify your results.
31. f(x) — x(x? - 3х + 3), g(x) — х?
32. y = x* – 2x², y = 2x²
33. f(x) = x4 – 4.x², g(x) = x² – 4
-
34. f(x) = x+ – 9x², g(x) = x³ – 9x
%3D
%3D
1
35. f(x)
1
g(x)
1 + x²'
6x
36. f(x) =
x2 + 1' y = 0, 0 < x< 3

Expert Solution

Step 1

Solution Q 33

Given,

$f (x) = x^{4} - 4 x^{2}$

$g (x) = x^{2} - 4$

Step 2

First finding the point of intersection

$f (x) = g (x)$

$x^{4} - 4 x^{2} = x^{2} - 4$

x⁴ - 5x² + 4 = 0

Let's assume t = x² and t² = x⁴

t² - 5t + 4 = 0

t² - 4t - t + 4 = 0

t(t-4)-1(t-4) = 0

(t-4)(t-1) = 0

t = 1 and t = 4

then

x² = 1 then x = $\pm 1$

x² = 4 then x = $\pm 2$

Step 3

Now sketching the graph of both the curve

Red curve represents $f (x) = x^{4} - 4 x^{2}$

blue curve represents $g (x) = x^{2} - 4$

Step by step

Solved in 5 steps with 1 images

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