Question

Use the Fundamental Theorem of Calculus to find the "area under curve" of y = − x 2 + 7 x between x = 1 and x = 5 . In your calculations, if you need to round, do not do so until the very end of the problem.

Answer 1

To find the area under the curve of y = -x² + 7x between x = 1 and x = 5 using the Fundamental Theorem of Calculus, evaluate the definite integral of the function over the interval. Find the antiderivative of the function, evaluate it at the upper and lower limits of integration, and subtract the results.

Step-by-step explanation:

To find the area under the curve of y = -x² + 7x between x = 1 and x = 5 using the Fundamental Theorem of Calculus, we need to evaluate the definite integral of the function over the given interval. The Fundamental Theorem of Calculus states that if F(x) is the antiderivative of f(x), then the definite integral of f(x) from a to b is equal to F(b) - F(a). In this case, we can find the antiderivative of -x² + 7x, which is -(1/3)x³ + (7/2)x². Evaluating this expression at x = 5 and x = 1, and subtracting the results, we can find the area under the curve.

1. Find the antiderivative of the function: F(x) = -(1/3)x³ + (7/2)x²
2. Evaluate the antiderivative at the upper limit of integration: F(5) = -(1/3)(5)³ + (7/2)(5)²
3. Evaluate the antiderivative at the lower limit of integration: F(1) = -(1/3)(1)³ + (7/2)(1)²
4. Subtract the results: F(5) - F(1) = [-(1/3)(5)³ + (7/2)(5)²] - [-(1/3)(1)³ + (7/2)(1)²]
5. Simplify and round the result if necessary

Answer 2

The area under the curve is of 42.67 area units.

Step-by-step explanation:

Using the Fundamental Theorem of Calculus, the area under a curve f(x), between
`x = a` and
`x = b`, is given by:

`A = \int_(a)^(b) f(x) dx`

In this question, we have that:

`f(x) = -x^2 + 7x`, from 1 to 5. So

`A = \int_(a)^(b) f(x) dx`

`A = \int_(1)^(5) (-x^2 + 7x) dx`

`A = -(x^3)/(3) + (7x^2)/(2)|_1^(5)`

`A = -(5^3)/(3) + (7*5^2)/(2) + (1^3)/(3) - (7*1^2)/(2)`

`A = -(124)/(3) + (168)/(2)`

`A = -(124)/(3) + 84`

`A = -(124)/(3) + (252)/(3)`

`A = (128)/(3)`

`A = 42.67`

The area under the curve is of 42.67 area units.