Definite Integral Quiz: Mastering Properties and Formulas

Study Notes

Integrals

Integrals are fundamental concepts in calculus, which is the branch of mathematics primarily concerned with rates of change and accumulation of quantities. There are two main types of integrals: definite integrals and indefinite integrals. In this article, we will focus on definite integrals.

Definite Integrals

A definite integral represents the signed area between the curve and/or its asymptotes. The notation for a definite integral indicates the limits of integration, such as:

∫(f(x) dx) from a to b = F(b) - F(a), where F(x) is the antiderivative of f(x).

The function symbol above the integral sign indicates the function whose antiderivative is sought. The limits of integration given as the bounds on x are represented by 'a' and 'b', where x = a and x = b. This notation emphasizes that the definite integral represents an exact quantity of signed area between two points.

Properties of Definite Integrals

Some properties of definite integrals include:

Additivity: If the function f(x) + g(x) has the same antiderivative as f(x) + h(x), then: ∫(f(x) dx) from a to b + ∫(h(x) dx) from a to b = ∫(g(x) dx) from a to b.
Linearity over a constant interval: If f(x) and g(x) have the same antiderivatives, then: ∫((af(x) + bg(x)) dx) from a to b = af(b) - af(a) + bg(b) - bg(a).
Fundamental Theorem of Calculus: If f(x) is continuous on [a, b], then there exists a function F(x) such that F'(x) = f(x) for all x in [a, b]. In other words, every continuous function has an antiderivative.

Examples of Definite Integrals

Let's look at some examples of calculating definite integrals:

Example 1: Find the value of the definite integral:

∫(-3x^2 + 3x - 2) dx from 1 to 2

To find this, we need to find the antiderivative of the function (-3x^2 + 3x - 2):

∫(-3x^2 dx) = (-3/3)x^(-2 + 1) = -x^(-1) = -1/(x^1) = -1/x.
∫(3x dx) = 3x(1/2) = 3/2x.
∫(-2 dx) = -2x.

So, the antiderivative of -3x^2 + 3x - 2 is:

(-1/x) + (3/2)x - 2x.

Now, to evaluate the definite integral from 1 to 2, we substitute x = 1 into the expression:

(-1/1) + (3/2)(1) - 2(1) = -1 + (3/2) - 2 = -1 - (3/2).

Therefore, the value of the definite integral is -1 - (3/2) = -2 - (3/2) = -(7/2).

Example 2: Evaluate the definite integral:

∫(e^x dx) from 1 to 2

To find this, we need to find the antiderivative of e^x:

∫(e^x dx) = e^x.

Now, to evaluate the definite integral from 1 to 2, we substitute x = 1 and x = 2 into the expression:

e^(1) - e^(2).

This gives us:

2.71828 - 7.38905 = -4.67077.

Therefore, the value of the definite integral is -4.67077.

Example 3: Find the definite integral:

∫((x^2 + 3)^3 dx) from 0 to 2

To find this, we need to find the antiderivative of (x^2 + 3)^3:

∫((x^2 + 3)^3 dx) = ((x^2 + 3)^4)/4.

Now, we evaluate the definite integral from 0 to 2:

(((2^2 + 3)^4)/4) - (((0^2 + 3)^4)/4) = ((16^4)/4) - ((0^4)/4) = 16^4 - 0 = 65536.

Therefore, the value of the definite integral is 65536.

Conclusion

Definite integrals represent the signed area between a curve and its asymptotes, with the limits of integration indicating the bounds on the x-axis. They have important properties, such as additivity and linearity, and can be evaluated using the fundamental theorem of calculus. By understanding these properties and examples, you can effectively work with definite integrals in various mathematical contexts.

Description

Explore the concept of definite integrals that represent the signed area between curves with specified limits of integration. Learn about properties like additivity, linearity, and how to evaluate definite integrals using the fundamental theorem of calculus. Practice examples to enhance your understanding of working with definite integrals.