Explanation Given information: The two curves has a function of y = x ( x 2 + 1 ) 2 and y = x 5 − x . Calculation: Procedure to sketch the region bounded by the two curves is explained below: Draw the graph for the function y = x ( x 2 + 1 ) 2 by substituting different values for x . Similarly in the same graph plot for the function y = x 5 − x by substituting different values for x . The region enclosed by the curves y = x ( x 2 + 1 ) 2 and y = x 5 − x is shown in Figure 1. Refer to Figure 1. The curves intersects at x = 0 and x = 1.052 . The curves are bounded by the top and bottom curve. Hence, the integration can be done with respect to x . Therefore, the point of intersection of the curves are at x = 0 _ and x = 1.052 _ . Find the area of the region bounded by the curves using the relation: A = ∫ a b ( f ( x ) − g ( x ) ) d x (1) Here, the top curve is f ( x ) , the bottom curve is g ( x ) , the lower limit is a , and the upper limit is b . Substitute x ( x 2 + 1 ) 2 for f ( x ) , x 5 − x for g ( x ) , 0 for a , and 1.052 for b in Equation (1). A = ∫ 0 1.052 [ ( x ( x 2 + 1 ) 2 ) − ( x 5 − x ) ] d x = ∫ 0 1.052 [ x ( x 2 + 1 ) 2 ] d x − ∫ 0 1.052 [ x 5 − x ] d x (2) Consider u = x 2 + 1 (3) Differentiate Equation (3) with respect to x . d u = 2 x x = d u 2 Find the value of the integral ∫ 0 1.052 [ x ( x 2 + 1 ) 2 ] d x : Substitute u for x 2 + 1 and d u 2 for x in the integral function ∫ 0 1.052 [ x ( x 2 + 1 ) 2 ] d x

9th Edition

ISBN: 9781337613927

Author: Stewart

Publisher: CENGAGE L

See similar textbooks

Concept explainers

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence t…

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

Videos

Question

Chapter 6.1, Problem 46E

To determine

To draw: The region enclosed by the given curves.

The point of intersection of the curves.

The area of the region enclosed by the curves.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.1, Problem 45E

Chapter 6.1, Problem 47E

Students have asked these similar questions

Question 1. (10 points) A researcher is studying tumours in mice. The growth rate for the volume of the tumour V(t) in cm³ is given by dV = 1.45V(2 In(V+1)). dt (a) (4 pts) Find all the equilibria and determine their stability using the stability condition. (b) (2 pts) Draw the phase plot f(V) versus V where f(V) = V'. You may find it helpful to use Desmos or Wolfram Alpha to plot the graph of f(V) versus V (both are free to use online), or you can plot it by hand if you like. On the plot identify each equilibrium as stable or unstable. (c) (4 pts) Draw direction arrows for the case where the tumour starts at size 3cm³ and for the case where the tumour starts at size 9cm³. Explain in biological terms what happens to the size of each of these tumours at time progresses.

For the system consisting of the two planes:plane 1: -x + y + z = 0plane 2: 3x + y + 3z = 0a) Are the planes parallel and/or coincident? Justify your answer. What does this tell you about the solution to the system?b) Solve the system (if possible). Show a complete solution. If there is a line of intersection express it in parametric form.

Question 2: (10 points) Evaluate the definite integral. Use the following form of the definition of the integral to evaluate the integral: Theorem: Iff is integrable on [a, b], then where Ax = (ba)/n and x₂ = a + i^x. You might need the following formulas. IM³ L² (3x² (3x²+2x- 2x - 1)dx. n [f(z)dz lim f(x)Az a n→∞ i=1 n(n + 1) 2 n i=1 n(n+1)(2n+1) 6

Chapter 6 Solutions

Calculus, Early Transcendentals

Ch. 6.1 - (a) Set up an integral for the area of the shaded...Ch. 6.1 - (a) Set up an integral for the area of the shaded...Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region. 5.Ch. 6.1 - Find the area of the shaded region. 6.Ch. 6.1 - Set up, but do not evaluate, an integral...Ch. 6.1 - Prob. 8E Ch. 6.1 - Set up, but do not evaluate, an integral...Ch. 6.1 - Set up, but do not evaluate, an integral...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To draw: The region enclosed by the given curves. The point of intersection of the curves. The area of the region enclosed by the curves.

Solution Summary: The procedure to sketch the region bounded by the curves is explained below.