Textbook Question
Volumes without calculus Solve the following problems with and without calculus. A good picture helps.
b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?
9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
2:07 minutesProblem 6.5.9
Textbook Question
9–20. Arc length calculations Find the arc length of the following curves on the given interval.
y = −8x−3 on [−2, 6] (Use calculus.)
Verified step by step guidance1
Identify the function given: $y = -8x - 3$ and the interval $[-2, 6]$ over which we want to find the arc length.
Recall the formula for the arc length $L$ of a curve $y = f(x)$ from $x = a$ to $x = b$:
$L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$
Compute the derivative $\frac{dy}{dx}$ of the function $y = -8x - 3$. Since this is a linear function, find $\frac{dy}{dx}$ explicitly.
Substitute $\frac{dy}{dx}$ into the arc length formula to get the integrand $\sqrt{1 + \left(\frac{dy}{dx}\right)^2}$, which will be a constant expression because the derivative is constant.
Evaluate the definite integral $\int_{-2}^{6} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$ by integrating the constant integrand over the interval length $(6 - (-2))$.
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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 sums the lengths of infinitesimal line segments along the curve, providing the total distance traveled along it.
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Derivative of the Function
To apply the arc length formula, you need the derivative dy/dx of the function y = f(x). The derivative represents the slope of the curve at any point, and its square is used inside the square root to account for the curve's steepness in the length calculation.
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Definite Integration
Calculating arc length requires evaluating a definite integral over the given interval [a, b]. This process involves integrating the expression √(1 + (dy/dx)^2) with respect to x, which may require algebraic simplification or substitution to solve.
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