Evaluate the integral using the following values.
8
∫ x^3 dx = 1020
2
8
∫ x dx = 30
2
8
∫ dx = 6
2
2
∫ x^3 dx = ?
2
To evaluate the integral ∫ x^3 dx from 2 to 8, you can use the power rule for integration. The power rule states that for any real number n (except -1), the integral of x^n dx is equal to x^(n+1)/(n+1) + C, where C is the constant of integration.
So, let's apply the power rule to the integral:
∫ x^3 dx = (x^4)/4 + C
To evaluate the definite integral from 2 to 8, plug in the upper limit (8) and the lower limit (2) into the antiderivative equation:
∫ (x^3) dx from 2 to 8 = [(8^4)/4 + C] - [(2^4)/4 + C]
Simplifying this expression:
= (4096/4 + C) - (16/4 + C)
= 1024 + C - 4 - C
= 1020
Therefore, ∫ (x^3) dx from 2 to 8 is equal to 1020.
To evaluate the integral ∫ x^3 dx with the given values of 2 and 8, we can use the power rule of integration. The power rule states that if we have an integral of the form ∫ x^n dx, where n is a constant, we can evaluate it by increasing the exponent by 1 and dividing by the new exponent.
In this case, we have ∫ x^3 dx. According to the power rule, we increase the exponent by 1 to get x^4, and then divide by the new exponent. So, the integral becomes ∫ x^3 dx = (1/4) * x^4.
Now, to calculate the definite integral with the given limits of 2 and 8, we substitute these values into the expression.
Substituting the upper limit 8, we get (1/4) * 8^4 = (1/4) * 4096 = 1024.
Then, substituting the lower limit 2, we get (1/4) * 2^4 = (1/4) * 16 = 4.
Finally, to find the value of the definite integral, we subtract the result of the lower limit from the result of the upper limit. So, 1024 - 4 = 1020.
Therefore, ∫ x^3 dx with limits 2 and 8 is equal to 1020.
Looks like you have the answers correct.
For the last one, recall that
∫[a,b] f(x) dx = F(b)-F(a)
So,
∫[a,a] f(x) dx = F(a)-F(a) = 0