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Calculate the area under the curve

y = x^3

from

x = 1

x = 4

Apr 30, 2024

Calculate the area under the curve

y = x^3

from

x = 1

x = 4

Generated Graph

Solution by Steps

step 1

To find the area under the curve

y = x^3

from

x = 1

x = 4

, we need to evaluate the definite integral

\int_{1}^{4} x^3 \, dx

step 2

The antiderivative of

x^3

\frac{x^4}{4}

step 3

We evaluate the antiderivative at the upper and lower limits of integration and subtract:

\left[\frac{x^4}{4}\right]_{1}^{4} = \frac{4^4}{4} - \frac{1^4}{4}

step 4

Simplifying the expression gives us

\frac{256}{4} - \frac{1}{4} = 64 - \frac{1}{4}

step 5

The final result is

64 - \frac{1}{4} = \frac{256}{4} - \frac{1}{4} = \frac{255}{4}

Answer

\frac{255}{4} = 63.75

Key Concept

Definite Integral

Explanation

The area under the curve from

x = 1

x = 4

is found by evaluating the definite integral of the function

x^3

over the interval [1, 4].