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Calculate the area under the curve

y = x^3

from

x = 1

x = 4

Nov 24, 2023

Calculate the area under the curve

y = x^3

from

x = 1

x = 4

Here is a Step-by-Step solution: #step 1#

Set up the definite integral for the area under the curve from

x = 1

x = 4

#step 2#

The integral is

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

#step 3#

Find the antiderivative of

x^3

, which is

\frac{x^4}{4}

#step 4#

Evaluate the antiderivative from

x = 1

x = 4

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

#step 5#

Substitute the limits of integration:

\frac{4^4}{4} - \frac{1^4}{4}

#step 6#

Calculate the result:

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

#step 7#

Simplify the expression:

64 - \frac{1}{4}

#step 8#

Final calculation:

64 - \frac{1}{4} = 63.75

Answer

The area under the curve

y = x^3

from

x = 1

x = 4

63.75

square units.

Key Concept

The definite integral of a function from a to b gives the area under the curve of the function between x = a and x = b.

Explanation

⚹By integrating the function

y = x^3

from

x = 1

x = 4

, we calculated the area under the curve. The antiderivative of

x^3

\frac{x^4}{4}

, and evaluating this from

x = 1

x = 4

gives us the area, which is

63.75

square units.