Determine the area of the shaded region in the following figur...

Question

Determine the area of the shaded region in the following figures.

Graph showing the intersection of y=2^x and y=3-x with a shaded area representing the region to calculate.

(Hint: Find the intersection point by inspection.)

Accepted Answer

Welcome back, everyone. Find the area enclosed by the shaded region in the figure below. A 3 halves minus 2 divided by LN of 3. B 7 halves minus 2 divided by LN of 3. C 7 halves plus 2 divided by LN of 3, and D 2 divided by L of 3 minus 7 halves. For this problem, let's identify the intersection points which define our shaded region. We notice that our shaded region begins at X equals 0, right? or basically at the origin. And then we have an intersection point at X equals 1. This is where the shaded region ends. So our limits of integration are going to be from 0 to 1. And we want to integrate the difference between two functions. Let's recall that. First of all, we take the upper curve. The upper curve that defines our region is the blue line which is 4 minus X. And we have to subtract the lower curve. The lower curve is the red one. It is below the blue one, right? And its equation is 3 raises to the power of X. This essentially means that our area is equal to the integral from 0 to 1 of 4 minus X, minus 3 raises to the power of XD X. Well then, so we have the expression of the area, and now we can integrate the integral of 4DX 4 X. The integral of negative axis x squared divided by 2 according to the power rule. In the integral of -3 to the power of X is negative 3 of X divided by LN of 3, using a table of integrals, right? And what we want to do is simply evaluate the result from 0 to 1. This is equal to 4 multiplied by 1 minus 1 squad divided by 2 minus 3 raises to the power of 1 divided by LN of 3. And now let's subtract the value of this function at X equals 0. So 4 multiplied by 0, minus 0 squad divided by 2, minus 3 to the power of 0 divided by L1 of 3. Let's simplify. We get 4 minus 1/2. Minus 3 divided by LN of 3, minus 00, and finally we're going to add. 1 divided by LN of 3, right? Because we have minus minus, which turns into a positive sign, and 3 to the power of 0 is 1. Let's combine the like terms, for it can be written as 8 divided by 2, and 8 minus 1 is 7. So we get 7 halves minus. -3 + 1 is -2, so we get -2 divided by LN of 3. So the correct answer to this problem is B 7/2s minus. 2 divided by Ellen of 3. Thank you for watching.

Determine the area of the shaded region in the following figures.

(Hint: Find the intersection point by inspection.)

Key Concepts

Intersection of Functions

Area Under a Curve

Definite Integral

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