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

Question

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

Accepted Answer

Welcome back, everyone. Find the area enclosed by the shaded region in the figure below. A 65 divided by 7, B 125 divided by 6, C 55 divided by 3, and D 25 divided by 6. So let's recall how we identify the area enclosed by two graphs. We want to take the integral. The lower limit corresponds to the first intersection point, and that would be X equals -3. And the upper limit of integration corresponds to the final intersection point, the second one, that will be X equals 2. So we're going to integrate from -3 up to 2, the difference between two functions. We have to take our upper function, that will be the red line, which is labeled as Y equals negative X. And we want to subtract the lower curve, which is the blue one. That's the parabola, right? It has the equation of x2 minus 6, meaning we're subtracting X2 minus 6, and we're integrating with respect to X. So we take the upper curve minus the lower curve. And now we are done setting up our integral, let's simplify. That's integral from -3 up to 2 of negative X minus X2 + 6DX. Now let's apply the power rules to integrate each term, the integral of negative axis, negative X squared divided by 2. We can also apply the power rule for the next term, we get negative X cubed divided by 3. And the integral of 6DX is simply 6 X. We want to evaluate the result from -3 up to 2. So let's apply the fundamental theorem of calculus part 2, we get negative 2 raise to the power of 2, so negative 2 squared divided by 2. Minus 2 cubed divided by 3 plus 6 multiplied by 2 and then we subtract the value of the function at -3 so negative. Of -3 2 divided by 2 minus. -3 cubed divided by 3. Plus 6 multiplied by -3. And now let's calculate the result. Now, 2 squared is 4, we get -4 divided by 2, which is -2. We then subtract 8/3, we add 12, so we're done with the first part, and now we subtract. Now, considering -3, we get -3 squad, which is 9, so we get -9 halves and we subtract negative halves. So this means we're basically adding 9 halves. Followed by -3 cubed. Now, -3 cubed is -27. We have a negative sign in front, so that's 27 divided by 3, which is 9, and then we subtract 9, right? And finally, we subtract -18, meaning we're basically adding 18. Now, let's combine the like terms. We have -2 + 12, which is 1010 minus 9 is 1, and 1 + 18 is 19. Constraining the fractional terms, we have negative 8/3 plus 9 halves, right? So the common denominator would be 6. So that would be minus 16 divided by 6. Plus 27 divided by 6. We can use our common denominator of 6. 9 multiplied by 6 is 114, we subtract 16 and we add 27. So now let's simplify. We get 6 as our denominator, 114 minus 16 is 98, and 98 plus 27 gives us 125. Well done, so we have our final answer, that's 125 divided by 6, which corresponds to the answer choice B. Thank you for watching.

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

Key Concepts

Definite Integral

Intersection Points

Area Between Curves

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