Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $ x $ and $ y $. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.

$ y = e^x $ , $ y = x^2 - 1 $ , $ x = -1 $ , $ x = 1 $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $ x $ and $ y $. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.

$ y = \sin x $ , $ y = x $ , $ x = \frac{\pi}{2} $ , $ x = \pi $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $ x $ and $ y $. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.

$ y = (x - 2)^2 $ , $ y = x $

Sky L.

Numerade Educator

$ y = x^2 - 4x $ , $ y = 2x $

Sky L.

Numerade Educator

$ y = \frac{1}{x} $ , $ y = \frac{1}{x^2} $ , $ x = 2 $

Sky L.

Numerade Educator

$ y = \sin x $ , $ y = \frac{2x}{\pi} $ , $ x \ge 0 $

Sky L.

Numerade Educator

$ x = 1 - y^2 $ , $ x = y^2 - 1 $

Sky L.

Numerade Educator

$ 4x + y^2 = 12 $ , $ x = y $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = 12 - x^2 $ , $ y = x^2 - 6 $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = x^2 $ , $ y = 4x - x^2 $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \sec^2 x $ , $ y = 8 \cos x $ , $ \frac{-\pi}{3} \le x \le \frac{\pi}{3} $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \cos x $ , $ y = 2 - \cos x $ , $ 0 \le x \le 2\pi $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ x = 2y^2 $ , $ x = 4 + y^2 $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \sqrt{x - 1} $ , $ x - y = 1 $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \cos \pi x $ , $ y = 4x^2 - 1 $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ x = y^4 $ , $ y = \sqrt{2 - 1} $ , $ y = 0 $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \tan x $ , $ y = 2 \sin x $ , $ \frac{-\pi}{3} \le x \le \frac{\pi}{3} $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = x ^3 $ , $ y = x $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \sqrt[3]{2x} $ , $ y = \frac{1}{8}x^2 $ , $ 0 \le x \le 6 $

Sky L.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \cos x $ , $ y = 1 - \cos x $ , $ 0 \le x \le \pi $

Apoorva S.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = x^4 $ , $ y = 2 - \mid x \mid $

Catherine R.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \sinh x $ , $ y = e^{-x} $ , $ x = 0 $ , $ x = 2 $

Kenneth K.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \frac{1}{x} $ , $ y = x $ , $ y = \frac{1}{4}x $ , $ x > 0 $

Kenneth K.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \frac{1}{4}x^2 $ , $ y = 2x^2 $ , $ x + y = 3 $ , $ x \ge 0 $

Kenneth K.

Numerade Educator

The graphs of two functions are shown with the areas of the regions between the curves indicated.

(a) What is the total area between the curves for $ 0 \le x \le 5 $?

(b) What is the value of $ \displaystyle \int_{0}^5 [f(x) - g(x)] dx $?

Kenneth K.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \frac{x}{\sqrt{1 + x^2}} $ , $ y = \frac{x}{\sqrt{9 - x^2}} $ , $ x \ge 0 $

Kenneth K.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \frac{x}{1 + x^2} $ , $ y = \frac{x^2}{1 + x^3} $

Kenneth K.

Numerade Educator

Sketch the region enclosed by the given curves and find its area.

$ y = \frac{\ln x}{x} $ , $ y = \frac{(\ln x)^2}{x} $

Kenneth K.

Numerade Educator

Use calculus to find the area of the triangle with the given vertices.

$ (0 , 0) $ , $ (3 , 1) $ , $ (1 , 2) $

Kenneth K.

Numerade Educator

Use calculus to find the area of the triangle with the given vertices.

$ (2 , 0) $ , $ (0 , 2) $ , $ (-1 , 1) $

Kenneth K.

Numerade Educator

Evaluate the integral and interpret it as the area of a region. Sketch the region.

$ \displaystyle \int_{0}^{\frac{\pi}{2}} \mid \sin x - \cos 2x \mid dx $

Kenneth K.

Numerade Educator

Evaluate the integral and interpret it as the area of a region. Sketch the region.

$ \displaystyle \int_{-1}^1 \mid 3^x - 2^x \mid dx $

Kenneth K.

Numerade Educator

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

$ y = x \sin (x^2) $ , $ y = x^4 $ , $ x \ge 0 $

Kenneth K.

Numerade Educator

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

$ y = \frac{x}{(x^2 + 1)^2} $ , $ y = x^5 - x $ , $ x \ge 0 $

Kenneth K.

Numerade Educator

Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

$ y = 3x^2 - 2x $ , $ y = x^3 - 3x + 4 $

Kenneth K.

Numerade Educator

$ y = 1.3^x $ , $ y = 2 \sqrt{x} $

Kenneth K.

Numerade Educator

Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

$ y = \frac{2}{1 + x^4} $ , $ y = x^2 $

Kenneth K.

Numerade Educator

Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

$ y = e^{1 - x^2} $ , $ y = x^4 $

Kenneth K.

Numerade Educator

Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

$ y = \tan^2 x $ , $ y = \sqrt{x} $

Kenneth K.

Numerade Educator

$ y = \cos x $ , $ y = x + 2 \sin^4 x $

Kenneth K.

Numerade Educator

Use a computer algebra system to find the exact area enclosed by the curves $ y = x^5 - 6x^3 + 4x $ and $ y = x $.

Kenneth K.

Numerade Educator

Sketch the region in the xy-plane defined by the inequalities $ x - 2y^2 \ge 0 $ , $ 1 - x - \mid y \mid \ge 0 $ and find its area.

Kenneth K.

Numerade Educator

Racing cars driven by Chris and Kelly are side by side at the start of a race. The table shows the velocities of each car (in miles per hour) during the first ten seconds of the race. Use the Midpoint Rule to estimate how much farther Kelly travels than Chris does during the first ten seconds.

Kenneth K.

Numerade Educator

The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool.

Kenneth K.

Numerade Educator

A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are 5.8, 20.3, 26.7, 29.0, 27.6, 27.3, 23.8, 20.5, 15.1, 8.7, and 2.8. Use the Midpoint Rule to estimate the area of the wing's cross section.

Kenneth K.

Numerade Educator

If the birth rate of a population is $ b(t) = 2200 e^{0.024t} $ people per year and the death rate is $ d(t) = 1460 e^{0.018t} $ people per year, find the area between these curves for $ 0 \le t \le 10 $. What does this area represent?

Bowen G.

Numerade Educator

In Example 5, we modeled a measles pathogenesis curve by a function $ f $. A patient infected with the measles virus who has some immunity to the virus has a pathogenesis curve that can be modeled by, for instance, $ g(t) = 0.9 f(t) $.

(a) If the same threshold concentration of the virus is required for infectiousness to begin as in Example 5, on what day does this occur?

(b) Let $ P_3 $ be the point of the graph of $ g $ where infectiousness begin. It has been shown that infectiousness ends at a point $ P_4 $ on the graph of $ g $ where the line through $ P_3 $, $ P_4 $ has the same slope as the line through $ P_1 $, $ P_2 $ in Example 5(b). On what day does infectiousness end?

(c) Compute the level of infectiousness for this patient.

Charles C.

Numerade Educator

The rates at which rain fell, in inches per hour, in two different locations $ t $ hours after the start of a storm are given by $ f(t) = 0.73t^3 - 2t^2 + t + 0.6 $ and $ g(t) = 0.17t^2 - 0.5t + 1.1 $. Compute the area between the graphs for $ 0 \le t \le 2 $ and interpret your result in this context.

Catherine R.

Numerade Educator

Two cars, A and B, start side by side and accelerate from rest. The figure shows the graphs of their velocity functions.

(a) Which car is ahead after one minute? Explain.

(b) What is the meaning of the area of the shaded region?

(c) Which car is ahead after two minutes? Explain.

(d) Estimate the time at which the cars are again side by side.

Charles C.

Numerade Educator

The figure shows graphs of the marginal revenue function $ R' $ and the marginal cost function $ C' $ for a manufacturer. [Recall from Section 4.7 that $ R(x) $ and $ C(x) $ represent the revenue and cost when $ x $ units are manufactured. Assume that $ R $ and $ C $ are measured in thousands of dollars.] What is the meaning of the area of the shaded region? Use the Midpoint Rule to estimate the value of this quantity.

Madi S.

Numerade Educator

The curve with equation $ y^2 = x^2 (x + 3) $ is called Tschirnhausen's cubic. If you graph this curve you will see that part of the curve forms a loop. Find the area enclosed by the loop.

Madi S.

Numerade Educator

Find the area of the region bounded by the parabola $ y = x^2 $, the tangent line to this parabola at $ (1, 1) $, and the x-axis.

Catherine R.

Numerade Educator

Find the number $ b $ such that the line $ y = b $ divides the region bounded by the curves $ y = x^2 $ and $ y = 4 $ into two regions with equal area.

Catherine R.

Numerade Educator

(a) Find the number a such that the line $ x = a $ bisects the area under the curve $ y = \frac{1}{x^2} $, $ 1 \le x \le 4 $.

Catherine R.

Numerade Educator

Find the values of $ c $ such that the area of the region bounded by the parabolas $ y = x^2 - c^2 $ and $ y = c^2 - x^2 $ is 576.

Catherine R.

Numerade Educator

Suppose that $ 0 < c < \frac{\pi}{2} $. For what value of $ c $ is the area of the region enclosed by the curves $ y = \cos x $, $ y = \cos (x - c) $, and $ x = 0 $ equal to the area of the region enclosed by the curves $ y = \cos (x - c) $, $ x = \pi $, and $ y = 0 $.

Madi S.

Numerade Educator

For what values of $ m $ do the line $ y = mx $ and the curve $ y = \frac{x}{(x^2 + 1)} $ enclose a region? Find the area of the region.

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