Question

1. Find the area of the region bounded above by y = e^x, bounded below by y = x, and bounded on the sides

x = 0 and x = 1

Answer 1

Using integrals, it is found that the area between the two curves is 0.412 units squared.

How to find the area between two curves with integrals?

The area between two curves, with f(x) being the upper curve and g(x) the lower curve, on an interval [a,b], is given by the following definite integral:

$\int_{a}^{b} (f(x) - g(x)) dx$

For this problem, the region we want to find the area is given at the graph at the end of the answer, hence, since the lower and upper bounds change, we have to use two integrals, as follows:

$A = A_1 + A_2$

$A_1 = \int_{0}^{0.567} (e^{-x} - x) dx$

$A_2 = \int_{0.567}^{1} (x - e^{-x}) dx$

Hence the first integral is:

$A_1 = (-e^{-x} - 0.5x^2)_{x = 0}^{x = 0.567}$

Applying the Fundamental Theorem of Calculus:

$A_1 = (-e^{-0.567} - 0.5(0.567)^2) -  (-e^{-0} - 0.5(0)^2)$

$A_1 = 0.272$

The second integral is:

$A_2 = (0.5x^2 + e^{-x})_{x = 0.567}^{x = 1}$

$A_2 = (0.5(1)^2 + e^{-1}) - (0.5(0.567)^2 + e^{-0.567})$

$A_2 = 0.14$

Hence the area between the curves is of:

$A = A_1 + A_2 = 0.272 + 0.14 = 0.412$

The area between the two curves is 0.412 units squared.

More can be learned about area and integrals at https://brainly.com/question/20733870

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