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Limits Of Integration

Limits of integration are used in definite integrals. The application of limits of integration to indefinite integrals transforms it into definite integrals. In the expression for integration ∫^a_b f(x).dx, for the function f(x), with limits [a, b], a is the upper limit and b is the lower limit. The limits of integration are applied in two steps: First, the integration of the function gives its antiderivative, and then limits are applied to the antiderivative of the function.

\(\int^a_b f(x).dx = [F(x)]^a_b = F(a) - F(b) \)

Let us learn more about how to solve limits of integration, formulas of limits of integration, with the help of examples, FAQs.

1.	What Are The Limits Of Integration?
2.	How To Solve Limits Of Integration?
3.	Formulas Of Limits Of Integration
4.	Examples On Limits Of Integration
5.	Practice Questions
6.	FAQs On Limits Of Integration

What Are The Limits Of Integration?

Limits of integration are the upper and the lower limits, which are applied to integrals. The integration of a function \(\int f(x)\) gives its antiderivative F(x), and the limits of integration [a, b] are applied to F(x), to obtain F(a) - F(b). Here in the given interval [a, b], a is called the upper limit and b is called the lower limit.

\(\int^a_b f(x).dx = [F(x)]^a_b = F(a) - F(b) \)

The area enclosed by the function across the bounding values, is found by integrating the function and applying the limits of integration. The upper limit and lower limit are the limits, which help to calculate the area enclosed by the curve. The integration involving limits of integration is called definite integrals. The final answer on applying limits of integration to the integral expression is a simple numeric value. The application of limits of integration to the function f(x), does not have any constant of integration, in the final answer.

How To Solve Limits Of Integration?

The limits of integration are solved across two steps. First, the integration is solved and then the limits of integration are applied. On applying the limits of integration two values of the function are obtained. The difference between the two values gives the final value of the limits of integration.

Step - I: The basic integration of the function f(x) with limits [a. b] gives the antiderivative of the function. \(\int^a_b f(x).dx = [F(x)]^a_b \)
Step - II: Next we apply the limits of integration[a, b] to the antiderivative answer F(x), to obtain the final answer. \([F(x)]^a_b = F(a) - F(b) \)

Thus the limits of integration are used to find a simple numeric value of the given integral expression.

Formulas Of Limits Of Integration

The following important formulas with limits of integration are used to find the final answer of definite integrals. Here the formulas of definite integrals are helpful to integrate the given function and apply the lower and the upper limit to find the value of the integral.

\(\int ^b_a f(x) .dx = \int^b _a f(t).dt \)
\(\int ^b_a f(x).dx = - \int^a _b f(x).dx \)
\(\int ^b_a cf(x).dx = c \int^b _a f(x).dx \)
\(\int ^b_a f(x) \pm g(x).dx = \int^b _a f(x).dx \pm \int^b_ag(x).dx\)
\(\int ^b_a f(x) .dx = \int^c _a f(x).dx + \int^b_cf(x).dx\)
\(\int ^b_a f(x) .dx = \int^b _a f(a + b - x).dx \)
\(\int ^a_0 f(x) .dx = \int^a _0 f(a - x).dx \) (This is a formula derived from the above formula.)
\(\int^{2a}_0f(x).dx = 2\int^a_0f(x).dx\) if f(2a - x) = f(x)
\(\int^{2a}_0f(x).dx = 0\) if f(2a - x) = -f(x).
\(\int^a_{-a}f(x).dx = 2\int^a_0f(x).dx\) if f(x) is an even function, and f(-x) = f(x).
\(\int^a_{-a}f(x).dx = 0\) if f(x) is an odd function, and f(-x) = -f(x).

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Examples on Limits Of Integration

Example 1: Find the integral by applying the limits of integration \(\int^3_{-3}x^5.dx\).

Solution:

\(\begin{align}\int^3_{-3}x^4.dx &= 2\int^3_0x^4.dx\\&=2\left[\frac{x^5}{5}\right]^3_0\\&=2(\frac{3^5}{5} - 0)\\&=2 \times \frac{243}{5}\\&=\frac{486}{5}\end{align}\)

Note: Here the function x⁴ is an even function, and hence we have used the formula \(\int^a_{-a}f(x).dx = 2\int^a_0f(x).dx\).

Answer: \(\int^3_{-3}x^5.dx =\frac{486}{5}\)
Example 2: Find the value of the integral \(\int^{\frac{\pi}{4}}_{\frac{-\pi}{4}} Sin^2x.dx\), by applying the limits of integration.

Solution:

\(\begin{align}\int^{\frac{\pi}{4}}_{\frac{-\pi}{4}} Sin^2x.dx &=2\int^{\frac{\pi}{4}}_0 Sin^2x.dx \\&=2\int^{\frac{\pi}{4}}_0 \dfrac{1 - Cos2x}{2}.dx\\&=\int^{\frac{\pi}{4}}_0(1 - cos2x).dx\\&=\left[x - \frac{Sin2x}{2}\right]^{\frac{\pi}{4}}_0\\&=\left(\frac{\pi}{4} - \frac{Sin\frac{\pi}{2}}{2}\right) - 0\\&=\frac{\pi}{4} - \frac{1}{2}\end{align}\)
Answer: \(\int^{\frac{\pi}{4}}_{\frac{-\pi}{4}} Sin^2x.dx=\frac{\pi}{4} - \frac{1}{2}\)

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Practice Questions on Limits Of Integration

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FAQs on Limits Of Integration

What Are the Limits Of Integration In Calculus?

The limits of integration are the upper and the lower boundaries which are applied to the integral function. The integration of a function \(\int f(x)\) gives its antiderivative F(x), and the limits of integration [a, b] are applied to F(x), to obtain F(a) - F(b). Here in the given interval [a, b], a is called the upper limit and b is called the lower limit.

\(\int^a_b f(x).dx = [F(x)]^a_b = F(a) - F(b) \)

How To Find The Limits Of Integration?

The limits of integration is generally given before the start of the integral function. The limits of integration for the function f(x) is \(\int^a_b f(x).dx\) and here a is the upper limit and b is the lower limit. The limits of integration are further applied to the solution o the integrals to find the final numeric value.

What Are The Formulas Of Limits Of Integration?

The formula for limits of integration is \(\int^a_b f(x).dx = [F(x)]^a_b = F(a) - F(b) \). Here the integral of the function f(x) is taken to obtain the antiderivative function F(x). Further the limits [a, b] are applied as the upper bound and the lower bound, and the difference of the function value is taken to find the final answer.

What Do You Call The Integration With Limits Of Integration?

The integration process involving the limits of integration are called definite integrals. The integration without any limits are referred as indefinite integrals.

What Are The Uses Of Limits Of Integration?

The limits of integration helps in finding the area enclosed by the curve within the bounding values. The limits of integration helps in finding the area enclosed by the function. The integration of the function f(x) gives the anti-derivative of the function, and further the upper bound and the lower bound given by the limits of integration, are applied to find the area enclosed by the curve.

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