Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.2.34a

Chapter 6, Problem 6.2.34a

For the given regions R₁ and R₂, complete the following steps.

a. Find the area of region R₁.

R₁ is the region in the first quadrant bounded by the y-axis and the curves y=2x^2 and y=3−x; R₂ is the region in the first quadrant bounded by the x-axis and the curves y=2x^2 and y=3−x(see figure).

Verified step by step guidance

Step 1: Identify the boundaries of region R₁. Region R₁ is bounded by the y-axis (x = 0), the curve y = 2x², and the line y = 3 − x. To find the area, we need to determine the points of intersection between y = 2x² and y = 3 − x.

Step 2: Solve for the points of intersection by equating the two equations: 2x² = 3 − x. Rearrange this into a standard quadratic form: 2x² + x − 3 = 0. Use the quadratic formula or factorization to find the values of x where the curves intersect.

Step 3: Set up the integral to calculate the area of R₁. The area is given by the integral of the top curve minus the bottom curve over the interval determined by the points of intersection. Specifically, the integral is: ∫[0, x₁] ((3 − x) − (2x²)) dx, where x₁ is the x-coordinate of the intersection point.

Step 4: Break down the integral into simpler components. Expand the integrand: (3 − x − 2x²). Compute the integral term by term: ∫[0, x₁] 3 dx, ∫[0, x₁] −x dx, and ∫[0, x₁] −2x² dx.

Step 5: Evaluate each integral over the interval [0, x₁] and combine the results to find the total area of region R₁. This will give the area enclosed by the curves and the y-axis.

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Key Concepts

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

Definite Integrals

Definite integrals are used to calculate the area under a curve between two points on the x-axis. In this context, the area of region R₁ can be found by integrating the difference between the upper curve (y = 3 - x) and the lower curve (y = 2x²) over the interval defined by their intersection points. This process quantifies the total area enclosed by the curves.

Finding Intersection Points

To determine the area of region R₁, it is essential to find the intersection points of the curves y = 2x² and y = 3 - x. These points are where the two curves meet, and they define the limits of integration for calculating the area. Solving the equation 2x² = 3 - x will yield the x-values at which the curves intersect.

Area Between Curves

The area between two curves is calculated by integrating the difference of the functions that define the curves over a specified interval. For region R₁, the area can be expressed as the integral of (3 - x - 2x²) dx, evaluated from the left intersection point to the right intersection point. This method effectively captures the space between the two curves.

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Related Practice

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Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).

a. If the tank is full, how much work is required to pump the water to the level of the top of the tank and out of the tank?

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Textbook Question

Flow rates in the Spokane River The daily discharge of the Spokane River as it flows through Spokane, Washington, in April and June is modeled by the functions

r1(t) = 0.25t²+37.46t+722.47 (April) and

r2(t) = 0.90t²−69.06t+2053.12 (June), where the discharge is measured in millions of cubic feet per day, and t=0 corresponds to the beginning of the first day of the month (see figure).

a. Determine the total amount of water that flows through Spokane in April (30 days).

Textbook Question

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.

a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

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Textbook Question

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.

a. Determine when the motion is in the positive direction and when it is in the negative direction.

v(t) = 50e^−2t on [0, 4]

Textbook Question

17–22. Position from velocity Consider an object moving along a line with the given velocity v and initial position.

a. Determine the position function, for t≥0, using the antiderivative method

v(t) = −t³+3t²−2t on [0, 3]; s(0)=4

Textbook Question

A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.

a. Use the shell method to write an integral for the volume of the torus.