Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.2.17

Chapter 6, Problem 6.2.17

Determine the area of the shaded region in the following figures.

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Identify the curves that bound the shaded region. The upper curve is the parabola y = x^2 - 4x, and the lower curve is the line y = -2x.

Find the points of intersection of the two curves by setting x^2 - 4x = -2x. Solve this equation to determine the x-coordinates of the intersection points.

Set up the integral to calculate the area between the curves. The area is given by the integral of the difference between the upper curve and the lower curve, ∫[(x^2 - 4x) - (-2x)] dx, over the interval determined by the intersection points.

Simplify the integrand to combine like terms. The integrand becomes x^2 - 2x, so the integral to evaluate is ∫(x^2 - 2x) dx over the interval.

Evaluate the definite integral by finding the antiderivative of x^2 - 2x, applying the Fundamental Theorem of Calculus, and substituting the limits of integration (the x-coordinates of the intersection points).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

A definite integral represents the signed area under a curve between two points on the x-axis. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. In this context, the area of the shaded region can be found by integrating the difference between the upper function (the parabola) and the lower function (the line) over the interval where they intersect.

Finding Intersection Points

To determine the area between two curves, it is essential to find their intersection points, as these points define the limits of integration. This involves solving the equations of the curves simultaneously to find the x-values where they intersect. In this case, solving the equations of the parabola and the line will yield the bounds for the definite integral.

Area Between Curves

The area between two curves is calculated by integrating the difference of the functions that define the curves over the interval determined by their intersection points. The formula for this area is given by ∫[a, b] (f(x) - g(x)) dx, where f(x) is the upper function and g(x) is the lower function. This concept is crucial for accurately determining the area of the shaded region in the provided graph.

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Finding Area Between Curves on a Given Interval

Related Practice

Textbook Question

29–36. Position and velocity from acceleration Find the position and velocity of an object moving along a straight line with the given acceleration, initial velocity, and initial position. Use the Fundamental Theorem of Calculus (Theorems 6.1 and 6.2).

a(t) = e^−t; v(0) = 60; s(0) = 40

Textbook Question

64–68. Shell method Use the shell method to find the volume of the following solids.

A hole of radius r≤R is drilled symmetrically along the axis of a bullet. The bullet is formed by revolving the parabola y = 6(1−x²/R²) about the y-axis, where 0≤x≤R.

Textbook Question

Lengths of symmetric curves Suppose a curve is described by y=f(x) on the interval [−b, b], where f′ is continuous on [−b, b]. Show that if f is odd or f is even, then the length of the curve y=f(x) from x=−b to x=b is twice the length of the curve from x=0 to x=b. Use a geometric argument and prove it using integration.

Textbook Question

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

y = -2

Textbook Question

Why is the disk method a special case of the general slicing method?

Textbook Question

Work from force How much work is required to move an object from x=1 to x=3 (measured in meters) in the presence of a force (in N) given by F(x) = 2x² acting along the x-axis?

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