Evaluate the following integral:
Integral lower limit 0 and upper limit 2
of 3x(x-7)dx
To evaluate the given integral, we can follow these steps:
1. Start by expanding the expression inside the integral:
3x(x - 7) = 3x^2 - 21x
2. Now, we can integrate each term separately:
∫(3x^2 - 21x) dx
∫(3x^2) dx = x^3
∫(-21x) dx = -10.5x^2
3. Apply the limits of integration:
We are integrating from 0 to 2, so we evaluate the antiderivatives at these limits:
(x^3 - 10.5x^2) evaluated from 0 to 2
4. Substitute the upper limit (2) into the antiderivatives:
(2^3 - 10.5(2)^2)
Simplifying:
(8 - 10.5(4))
(8 - 42)
-34
5. Substitute the lower limit (0) into the antiderivatives:
(0^3 - 10.5(0)^2)
Simplifying:
(0 - 10.5(0))
(0 - 0)
0
6. Finally, subtract the result of the lower limit from the result of the upper limit:
-34 - 0 = -34
Therefore, the value of the given integral ∫(3x(x - 7)) dx from 0 to 2 is -34.