Find the exact area under the curve between the indicated values of $x$ . $y = \frac{3}{x^{3}}$ , between $x = 1$ and $x = 3$
A. $\frac{1}{2}$
B. 3
C. $\frac{1}{3}$
D. $\frac{4}{3}$

Answer

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Answer

Final Answer: $\frac{4}{3}$

Steps

Step 1 ：We are given the function $y = \frac{3}{x^{3}}$ and we are asked to find the exact area under the curve between $x = 1$ and $x = 3$ .

Step 2 ：To find the exact area under the curve, we need to integrate the function from the lower limit to the upper limit.

Step 3 ：The integral of $\frac{3}{x^{3}}$ is $- \frac{3}{2 x^{2}}$ .

Step 4 ：We will evaluate this at $x = 3$ and $x = 1$ and subtract the two results to find the area.

Step 5 ：The result of the integration is $\frac{4}{3}$ , which is the exact area under the curve between $x = 1$ and $x = 3$ .

Step 6 ：Final Answer: $\frac{4}{3}$