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 6.2.71

Chapter 6, Problem 6.2.71

Determine the area of the shaded region bounded by the curve x^2=y^4(1−y^3) (see figure).

Verified step by step guidance

Step 1: Analyze the given equation x^2 = y^4(1 - y^3). The shaded region is symmetric about the y-axis, so we can calculate the area for y in [0, 1] and double it to account for symmetry.

Step 2: Express x in terms of y. Since x^2 = y^4(1 - y^3), the curve is symmetric about the y-axis, and we can focus on the positive x values. Thus, x = sqrt(y^4(1 - y^3)).

Step 3: Set up the integral for the area. The area of the shaded region is given by the integral of x with respect to y over the interval [0, 1]. The integral becomes A = 2 * ∫[0 to 1] sqrt(y^4(1 - y^3)) dy.

Step 4: Simplify the integrand. Rewrite sqrt(y^4(1 - y^3)) as y^2 * sqrt(1 - y^3) to make the integral easier to handle.

Step 5: Solve the integral. Use substitution to evaluate the integral. Let u = 1 - y^3, then du = -3y^2 dy. Adjust the limits of integration accordingly and proceed to compute the integral.

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Key Concepts

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

Implicit Functions

The equation x² = y⁴(1 - y³) defines a relationship between x and y, where y is expressed implicitly as a function of x. Understanding implicit functions is crucial for analyzing curves that cannot be easily solved for one variable in terms of another. This concept is foundational in calculus, particularly when dealing with curves and their properties.

Area Under a Curve

Calculating the area of a shaded region bounded by a curve involves integrating the function that describes the curve. In this case, the area can be found by setting up an integral with appropriate limits based on the intersection points of the curve with the axes. This concept is essential for understanding how to quantify regions in the Cartesian plane.

Symmetry in Graphs

The given curve exhibits symmetry about the y-axis, which can simplify the calculation of the area. Recognizing symmetry allows us to compute the area of one side of the curve and then double it, reducing the complexity of the problem. This concept is particularly useful in calculus when dealing with even functions or symmetric shapes.

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Related Practice

Textbook Question

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

Textbook Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x² and y = 2−x²; about the x-axis

Textbook Question

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis

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Textbook Question

Find the area of the region described in the following exercises.

The region bounded by y=2 / 1 + x^2 and y=1

Textbook Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=ln x,y=ln x^2; and y=ln 8; about the y-axis

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Textbook Question

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

(Hint: Find the intersection point by inspection.)