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yaohui's Question
Math
Posted 27 days ago
Calculate the area under the curve y=x3y = x^3 from x=1x = 1 to x=4x = 4.
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Answer from Sia
Posted 27 days ago
Solution by Steps
step 1
To find the area under the curve y=x3y = x^3 from x=1x = 1 to x=4x = 4, we need to evaluate the definite integral 14x3dx\int_{1}^{4} x^3 \, dx
step 2
The integral of x3x^3 is given by x3dx=x44+C\int x^3 \, dx = \frac{x^4}{4} + C
step 3
We now evaluate this antiderivative at the bounds x=4x = 4 and x=1x = 1: x4414=444144. \left. \frac{x^4}{4} \right|_{1}^{4} = \frac{4^4}{4} - \frac{1^4}{4}.
step 4
Simplifying the expression: 444=2564=64, \frac{4^4}{4} = \frac{256}{4} = 64, 144=14. \frac{1^4}{4} = \frac{1}{4}. Thus, the area is: 6414=256414=2554=63.75. 64 - \frac{1}{4} = \frac{256}{4} - \frac{1}{4} = \frac{255}{4} = 63.75.
Answer
The area under the curve y=x3y = x^3 from x=1x = 1 to x=4x = 4 is 63.7563.75.
Key Concept
Definite Integral
Explanation
The definite integral of a function over an interval gives the area under the curve of that function within the specified bounds.

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