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 6.2.34a

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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.

Recommended video:

05:23

Finding Area Between Curves on a Given Interval

Related Practice

Textbook Question

40–43. Population growth

A culture of bacteria in a Petri dish has an initial population of 1500 cells and grows at a rate (in cells/day) of N′(t) = 100e^−0.25t. Assume t is measured in days.

a. What is the population after 20 days? After 40 days?

views

Textbook Question

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?

154

views

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).

views

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

views

Textbook Question

Blood flow A typical human heart pumps 70 mL of blood (the stroke volume) with each beat. Assuming a heart rate of 60 beats/min (1 beat/s), a reasonable model for the outflow rate of the heart is V′(t)=70(1+sin 2πt), where V(t) is the amount of blood (in milliliters) pumped over the interval [0,t],V(0)=0 and t is measured in seconds.

a. Verify that the amount of blood pumped over a one-second interval is 70 mL.

162

views

Textbook Question

Region R is revolved about the line y=1 to form a solid of revolution.

a. What is the radius of a cross section of the solid at a point x in [0, 4]?

views