Textbook Question
A solid has a circular base; cross sections perpendicular to the base are squares. What method should be used to find the volume of the solid?
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.2.37
Find the area of the region described in the following exercises.
The region bounded by y=4x+4, y=6x+6, and x=4
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Step 1: Identify the boundaries of the region. The region is bounded by the lines y = 4x + 4, y = 6x + 6, and x = 4. These equations describe straight lines, and the intersection points will help define the vertices of the region.
Step 2: Find the intersection points of the lines. To find where y = 4x + 4 intersects y = 6x + 6, set the two equations equal to each other: . Solve for x, then substitute the x-value into either equation to find the corresponding y-coordinate.
Step 3: Determine the points where the lines intersect the vertical line x = 4. Substitute x = 4 into y = 4x + 4 and y = 6x + 6 to find the y-coordinates of the intersection points. This will give you two points on the vertical line.
Step 4: Sketch the region. Plot the lines y = 4x + 4, y = 6x + 6, and x = 4 on a graph. Mark the intersection points found in Steps 2 and 3. The region is a triangular area enclosed by these boundaries.
Step 5: Calculate the area of the triangular region. Use the formula for the area of a triangle: . Determine the base and height of the triangle from the graph or the coordinates of the vertices, and substitute these values into the formula.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Equations
Linear equations represent straight lines in a coordinate plane. The equations given, y=4x+4 and y=6x+6, describe two lines with different slopes. Understanding how to graph these lines is essential for visualizing the bounded region and determining the area between them.
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Linearization
Area Between Curves
The area between two curves can be found by integrating the difference of the functions over a specified interval. In this case, the area between the lines y=4x+4 and y=6x+6 from the leftmost intersection point to x=4 needs to be calculated. This involves setting up the integral of the upper function minus the lower function.
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Definite Integrals
Definite integrals are used to calculate the area under a curve between two points on the x-axis. They provide a numerical value representing the total area, which is crucial for solving the problem. The limits of integration will be determined by the points where the two lines intersect and the vertical line x=4.
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05:43Definition of the Definite Integral
Related Practice
Textbook Question
Force on the end of a tank Determine the force on a circular end of the tank in Figure 6.78 if the tank is full of gasoline. The density of gasoline is ρ = 737 kg/m³.
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=sin xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.
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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=x,y=x+2,x=0, and x=4 ; about the x-axis
Textbook Question
9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.
{Use of Tech} y = 1 / (x² + 1)²,y=0,x=1, and x=2; about the y-axis
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Textbook Question
Find the arc length of the line y = 4−3x on [−3, 2] using calculus and verify your answer using geometry.