Question

Find the area of the curve that lies under the parabola 5x - x^2 where 1 ≤ x ≤ 4

A. 6.33
B. 8.67
C. 10.21
D. 12.45

Answer 1

The area under the parabola 5x - x^2 from x=1 to x=4 is found through integration and results in 8.67 square units.

The question involves finding the area under a given parabola, which is a quadratic function, within specified limits. The parabolic equation provided is 5x - x2. To find the area under the curve from x=1 to x=4, we use integration over these bounds. The integral of 5x - x2 from 1 to 4 gives us the exact area under the curve.

Explanation:

1. Write the integral of the function 5x - x2 between the limits of 1 and 4.
2. Calculate the antiderivative: (5/2)x2 - (1/3)x3.
3. Evaluate this antiderivative from 1 to 4.
4. The final calculation yields the area under the curve.

The exact calculation is ((5/2)*(4)2 - (1/3)*(4)3) - ((5/2)*(1)2 - (1/3)*(1)3), which simplifies to 8.67.

In conclusion, the area under the curve of the parabola 5x - x2 from x=1 to x=4 is 8.67 square units, making option B the correct answer.