Definite Integrals Calculator

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Example

Created on 2024-06-20Asked by Avery Wright (Solvelet student)

Calculate

\int_{0}^{1} x^2 \, dx

Solution

Step 1: Identify the integrand

f(x) = x^2

and the limits of integration

a = 0

and

b = 1

. Step 2: Use the definite integral formula:

\int_{a}^{b} f(x) \, dx = F(b) - F(a),

where

F(x)

is the antiderivative of

f(x)

. Step 3: Find the antiderivative of

f(x)

F(x) = \frac{1}{3}x^3.

Step 4: Evaluate

F(b) - F(a)

\int_{0}^{1} x^2 \, dx = F(1) - F(0) = \frac{1}{3}(1)^3 - \frac{1}{3}(0)^3 = \frac{1}{3}.

Step 5: Conclusion.

\int_{0}^{1} x^2 \, dx = \frac{1}{3}

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Benjamin Miller on Solvelet

1. Evaluate the definite integral

\int_{0}^{\pi//2} \sin(x) \, dx

2. Calculate the area bounded by the curves

y = x^2

and

y = x + 2

DefinitionDefinite integrals: this is the de or integral type in which the points in the interval of the integral are set to well-deﬁned boundaries. The deﬁnite integral ∫abf x dx shows a net area of the region for the curve, in terms of f x and the x-axis. In calculus, a deﬁnite integral describes how the quantity and the rate of its changes are related. Change is deﬁned with respect to the distance of something to be covered over time, distance covered per head count, change in value with respect to quantity, a diﬀerence which varies along the positionPales of a second. For example, the deﬁnite integral ∫01x2dx gives the are under the curve y=x2 from x=0 to x=1 for the X-axis.

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