Question

Evaluate the following definite integral (to three decimal places):

1
∫ x² dx
0
A) 0.333
B) 0.250
C) 0.167
D) 0.125

Answer 1

The definite integral of x² from 0 to 1 is evaluated using the power rule of integration and results in a value of 1/3, or 0.333 when expressed as a decimal. The correct option is A) 0.333.

Step-by-step explanation:

The student is asking to evaluate the definite integral of the function f(x) = x² from the limits x = 0 to x = 1. To do this, we use the power rule of integration, which tells us that the integral of x^n dx, where n is any real number except -1, is (x^(n+1))/(n+1) + C, where C is the constant of integration. However, since this is a definite integral, we will not be including the constant of integration in our evaluation.

Applying the power rule, we get:

1. Integral of x² from 0 to 1 is (x^(2+1))/(2+1) from 0 to 1.
2. This simplifies to (x³)/3 from 0 to 1.
3. Now we substitute the upper limit into the expression and then subtract the expression with the lower limit substituted in. So it becomes (1³)/3 - (0³)/3.
4. This results in 1/3 - 0 which equals 1/3.

Therefore, the value of the definite integral is 1/3, which expressed as a decimal to three decimal places, is 0.333. Thus, the correct answer is Option A).