AskSia

Plus

Calculate the area under the curve

y = x^3

from

x = 1

x = 4

Nov 18, 2023

Calculate the area under the curve

y = x^3

from

x = 1

x = 4

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

To calculate the area under the curve

y = x^3

from

x = 1

x = 4

, we will use the definite integral of the function

y = x^3

over the interval

[1, 4]

#step 2#

Set up the definite integral for the function

y = x^3

from

x = 1

x = 4

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

#step 3#

Find the antiderivative of

x^3

. The antiderivative of

x^n

\frac{x^{n+1}}{n+1}

, so the antiderivative of

x^3

\frac{x^{4}}{4}

#step 4#

Evaluate the antiderivative at the upper and lower limits of the integral and subtract:

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

#step 5#

Simplify the expression to find the area under the curve:

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

#step 6#

The final result is the area under the curve from

x = 1

x = 4

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

Definite Integral

Explanation

The definite integral of a function over an interval gives the area under the curve of the function within that interval. In this case, the definite integral of

x^3

from

x = 1

x = 4

represents the area under the curve.