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Calculate the area under the curve

y = x^3

from

x = 1

x = 4

Feb 1, 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}

, so we use the Fundamental Theorem of Calculus to evaluate the integral

step 3

We substitute the upper and lower limits of the integral into the antiderivative:

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

step 4

Calculating the value at the upper limit:

\frac{4^4}{4} = \frac{256}{4} = 64

step 5

Calculating the value at the lower limit:

\frac{1^4}{4} = \frac{1}{4}

step 6

Subtracting the lower limit value from the upper limit value gives us the area under the curve:

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

Answer

\frac{255}{4}

63.75

Key Concept

Definite Integral to Find Area

Explanation

The area under a curve from one point to another is found by evaluating the definite integral of the function between those two points. The definite integral gives the net area, which is the area above the x-axis minus the area below the x-axis between the given limits.