Textbook Question
39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.
y = -2
9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
1:51 minutesProblem 6.5.38
Textbook Question
Function defined as an integral Write the integral that gives the length of the curve y = f(x) = ∫₀^x sin t dt on the interval [0,π]
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Recall the formula for the length of a curve defined by a function $y = f(x)$ on the interval $[a,b]$: the length $L$ is given by the integral $L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$.
Identify the function $f(x)$ given as an integral: $f(x) = \int_0^x \sin t \, dt$.
Find the derivative $\frac{dy}{dx}$ of the function $f(x)$ using the Fundamental Theorem of Calculus, which states that if $f(x) = \int_a^x g(t) \, dt$, then $f'(x) = g(x)$. So, $\frac{dy}{dx} = \sin x$.
Substitute $\frac{dy}{dx} = \sin x$ into the length formula to get $L = \int_0^{\pi} \sqrt{1 + (\sin x)^2} \, dx$.
Write the final integral expression for the length of the curve on $[0, \pi]$ as $L = \int_0^{\pi} \sqrt{1 + \sin^2 x} \, dx$.
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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 given by the integral ∫_a^b √(1 + (dy/dx)²) dx. This formula calculates the length by summing infinitesimal line segments along the curve, accounting for both horizontal and vertical changes.
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Fundamental Theorem of Calculus
This theorem connects differentiation and integration, stating that if f(x) = ∫_a^x g(t) dt, then f'(x) = g(x). It allows us to find the derivative of an integral-defined function, which is essential for computing dy/dx in the arc length formula.
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Derivative of the Given Function
Given y = ∫₀^x sin t dt, by the Fundamental Theorem of Calculus, dy/dx = sin x. Knowing this derivative is crucial to substitute into the arc length formula to express the integral that calculates the curve's length on [0, π].
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