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Write an integral expression that will give the length of the path given by
f(x)=10x^(4)-1 from
x=7 to
x=8.

Write an integral expression that will give the length of the path given by $f(x)=10x^{4}-1$ from $x=7$ to $x=8$ .

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Evaluate definite integrals using the chain rule

Full solution

Q. Write an integral expression that will give the length of the path given by

f(x)=10x^{4}-1

from

x=7

to

x=8

.

Find Derivative of f(x): To find the length of the path of a function $f(x)$ from $x = a$ to $x = b$ , we use the arc length formula, which is given by the integral: $\newline$ $L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx$ $\newline$ First, we need to find the derivative of $f(x) = 10x^4 - 1$ .
Substitute into Arc Length Formula: The derivative of $f(x) = 10x^4 - 1$ with respect to $x$ is $f'(x) = \frac{d}{dx} (10x^4 - 1) = 40x^3$ .
Simplify Integral Expression: Now we substitute $f'(x)$ into the arc length formula and simplify: $L = \int_{7}^{8} \sqrt{1 + (40x^3)^2} \, dx$
Simplify Integral Expression: Now we substitute $f'(x)$ into the arc length formula and simplify: $\newline$ $L = \int_{7}^{8} \sqrt{1 + (40x^3)^2} \, dx$ Simplify the expression inside the square root: $\newline$ $L = \int_{7}^{8} \sqrt{1 + 1600x^6} \, dx$ $\newline$ This is the integral expression that will give the length of the path of $f(x)$ from $x = 7$ to $x = 8$ .

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