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Calculate the area under the curve

y = x^3

from

x = 1

x = 4

Dec 19, 2023

Calculate the area under the curve

y = x^3

from

x = 1

x = 4

Solution by Steps

step 1

To calculate the area under the curve

y = x^3

from

x = 1

x = 4

, we need to find the definite integral of the function within these limits

step 2

The antiderivative of

x^3

is found using the power rule for integration:

\int x^n dx = \frac{x^{n+1}}{n+1} + C

, where

C

is the constant of integration

step 3

Applying the power rule to

x^3

, we get

\int x^3 dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C

step 4

We evaluate the antiderivative from

x = 1

x = 4

\left[ \frac{x^4}{4} \right]_1^4 = \frac{4^4}{4} - \frac{1^4}{4}

step 5

Simplifying the expression:

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

step 6

Subtracting the two values gives us the area under the curve:

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

Answer

The area under the curve

y = x^3

from

x = 1

x = 4

is 63.75 square units.

Key Concept

Definite Integration to Find Area Under a Curve

Explanation

The area under the curve

y = x^3

from

x = 1

x = 4

is found by evaluating the definite integral of the function between these limits, which gives the total accumulated area.