Textbook Question
Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).
b. How is this formula changed if x0>b?
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9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
3:39 minutesProblem 6.5.8
Textbook Question
Find the arc length of the line y = 4−3x on [−3, 2] using calculus and verify your answer using geometry.
Verified step by step guidance1
Recall the formula for the arc length of a curve defined by a function $y = f(x)$ on the interval $[a, b]$:
$$L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$
Identify the function and interval: here, $y = 4 - 3x$ and the interval is $[-3, 2]$.
Compute the derivative $\frac{dy}{dx}$: since $y = 4 - 3x$, then
$$\frac{dy}{dx} = -3$$
Substitute the derivative into the arc length formula:
$$L = \int_{-3}^{2} \sqrt{1 + (-3)^2} \, dx = \int_{-3}^{2} \sqrt{1 + 9} \, dx = \int_{-3}^{2} \sqrt{10} \, dx$$
Evaluate the integral: since $\sqrt{10}$ is constant, the integral becomes
$$L = \sqrt{10} \times (2 - (-3)) = \sqrt{10} \times 5$$.
To verify using geometry, recognize that the graph is a straight line segment between points $(-3, 4 - 3(-3))$ and $(2, 4 - 3(2))$. Calculate the distance between these points using the distance formula
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
and confirm it matches the arc length.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length Formula
The arc length of a curve y = f(x) from x = a to x = b is found using the integral formula L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the length of the curve by summing infinitesimal line segments along the curve.
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Derivative of a Function
The derivative dy/dx represents the slope of the function y = f(x) at any point x. For the arc length formula, the derivative is squared and added to 1 inside the square root to account for both horizontal and vertical changes along the curve.
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Geometric Interpretation of a Line Segment
Since y = 4 - 3x is a straight line, its arc length over [−3, 2] can be found using the distance formula between two points: √((x2 - x1)^2 + (y2 - y1)^2). This provides a geometric verification of the calculus result.
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