[Clean the question above and then proceed with classification]
$\int_{-8}^2 16-6 y-y^2$
Answer (1)
3 500
Explanation
Problem Setup We are asked to evaluate the definite integral: ∫ − 8 2 ( 16 − 6 y − y 2 ) d y To do this, we will first find the antiderivative of the integrand, then evaluate the antiderivative at the upper and lower limits of integration, and finally subtract the value at the lower limit from the value at the upper limit.
Finding the Antiderivative First, let's find the antiderivative of 16 − 6 y − y 2 . Using the power rule for integration, we have: ∫ ( 16 − 6 y − y 2 ) d y = 16 y − 3 y 2 − 3 1 y 3 + C where C is the constant of integration.
Evaluating at Limits Now, we evaluate the antiderivative at the upper limit of integration, which is 2: 16 ( 2 ) − 3 ( 2 ) 2 − 3 1 ( 2 ) 3 = 32 − 12 − 3 8 = 20 − 3 8 = 3 60 − 8 = 3 52 Next, we evaluate the antiderivative at the lower limit of integration, which is -8: 16 ( − 8 ) − 3 ( − 8 ) 2 − 3 1 ( − 8 ) 3 = − 128 − 3 ( 64 ) − 3 1 ( − 512 ) = − 128 − 192 + 3 512 = − 320 + 3 512 = 3 − 960 + 512 = 3 − 448
Calculating the Definite Integral Finally, we subtract the value of the antiderivative at the lower limit from the value at the upper limit: 3 52 − ( 3 − 448 ) = 3 52 + 3 448 = 3 52 + 448 = 3 500 So, the value of the definite integral is 3 500 .
Final Answer Therefore, the definite integral ∫ − 8 2 ( 16 − 6 y − y 2 ) d y is equal to 3 500 .
Examples
Definite integrals are used extensively in physics and engineering to calculate quantities such as the area under a curve, the work done by a force, or the total distance traveled by an object. For example, if you have a function that describes the velocity of a car over time, the definite integral of that function over a certain time interval will give you the total distance the car traveled during that time. This is a fundamental concept in understanding motion and other physical phenomena.
