Textbook Question
Find the area of the region described in the following exercises.
The region bounded by y=|x−3|and y=x/2
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9. Graphical Applications of Integrals
Area Between Curves
3:49 minutesProblem 6.2.29
Textbook Question
Determine the area of the shaded region in the following figures.
Verified step by step guidance1
Step 1: Identify the curves that bound the shaded region. From the graph, the shaded region is enclosed between the curves x = y^2 - 3y + 12 (black curve) and x = -2y^2 - 6y + 30 (red curve).
Step 2: Determine the limits of integration. The shaded region spans from y = -2 to y = 2, as indicated by the intersection points of the two curves along the y-axis.
Step 3: Set up the integral for the area. The area of the shaded region is given by the integral of the difference between the rightmost curve (x = -2y^2 - 6y + 30) and the leftmost curve (x = y^2 - 3y + 12) with respect to y. The integral is: ∫[-2 to 2] [(-2y^2 - 6y + 30) - (y^2 - 3y + 12)] dy.
Step 4: Simplify the integrand. Combine like terms to simplify the expression inside the integral: (-2y^2 - 6y + 30) - (y^2 - 3y + 12) = -3y^2 - 3y + 18.
Step 5: Evaluate the integral. Integrate the simplified expression -3y^2 - 3y + 18 with respect to y over the interval [-2, 2]. This involves finding the antiderivative and substituting the limits of integration.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral calculates the net area under a curve between two specified points on the x-axis. It is represented as ∫[a, b] f(x) dx, where f(x) is the function being integrated, and a and b are the limits of integration. This concept is crucial for finding the area of the shaded region between two curves, as it allows us to quantify the space enclosed by them.
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Definition of the Definite Integral
Intersection Points
Intersection points are the values of x where two curves meet, which are essential for determining the limits of integration in area calculations. To find these points, one must set the equations of the curves equal to each other and solve for x. Identifying these points helps in accurately defining the boundaries of the shaded region whose area we want to calculate.
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Area Between Curves
The area between two curves is found by integrating the difference of the functions that define the curves over the interval defined by their intersection points. If f(x) is the upper curve and g(x) is the lower curve, the area A can be expressed as A = ∫[a, b] (f(x) - g(x)) dx. This concept is fundamental for solving the problem of finding the area of the shaded region in the given figure.
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