`Calculate the area under the curve $y = x^3$ from $x = 1$ to $x = 4$.`

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Answer from Sia

Posted 27 days ago

Solution by Steps

step 1

To find the area under the curve $y = x^3$ from $x = 1$ to $x = 4$, we need to evaluate the definite integral $\int_{1}^{4} x^3 \, dx$

step 2

The integral of $x^3$ is given by $\int x^3 \, dx = \frac{x^4}{4} + C$

step 3

We now evaluate this antiderivative at the bounds $x = 4$ and $x = 1$:
$\left. \frac{x^4}{4} \right|_{1}^{4} = \frac{4^4}{4} - \frac{1^4}{4}.$

step 4

Simplifying the expression:
$\frac{4^4}{4} = \frac{256}{4} = 64,$
$\frac{1^4}{4} = \frac{1}{4}.$
Thus, the area is:
$64 - \frac{1}{4} = \frac{256}{4} - \frac{1}{4} = \frac{255}{4} = 63.75.$

Answer

The area under the curve $y = x^3$ from $x = 1$ to $x = 4$ is $63.75$.

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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