Use geometry and properties of integrals to evaluate the follo...

Question

Use geometry and properties of integrals to evaluate the following definite integrals.

∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍 . (Hint: Write the integral as sum of two integrals.)

Accepted Answer

Hello, in this video, we are going to be finding the value of the given definite integral by using geometric reasoning and the properties of the integrals. So the definite integral given to us is the integral from 3 to 0 of 4 X plus the square root of 9 minus X squared. Now, notice that the bounds are given to us in the order from largest bound to the smallest bound. Normally, we like to write our definite integrals with the smallest value, leading to the larger value. So what we are going to do is we are going to flip the bounds by taking the negative version of this integral. So we can rewrite the integral as the negative definite integral from 0 to 3 of 4 X plus the square root of 9 minus X squared. Next, what we are going to do is we are going to split this up into a sum of definite integrals. So we're going to have the negative quantity of the definite integral from 0 to 3 of 4 X. Plus the definite integral from 0 to 3 of the square root of 9 minus X 2 D X. So, let's go ahead and solve for the first integral. Now, the first integral, again, is the integral from 0 to 3 of 4 X. If we take the antiderivative of this integral, we will end up with 4 multiplied by 1/2 X squad, and this will be evaluated from 0 to 3. By multiplying the coefficients together, we will have 2 X2d evaluated from 0 to 3. Next, by using the fundamental theorem of calculus part 2, we will be left with 2 multiplied by 32 minus 2 multiplied by 02, and this is going to simplify to give us the value of 18. So this is going to be the value of the first integral. Now, what we want to do is we want to solve for the second integral, but let's go ahead and solve this one by using geometric reasoning. Now, this is the integral from 0 to 3 of the square root 9 minus X squared. Now, 9 minus X2d represents the upper half of a circle centered at 0 with the radius of 3, but in this case here, because we only want the positive upper half, this equation represents a quarter circle in the first quadrant. So the graph of this part of the integral is going to be a circle of radius 3. Located in the first quadrant of the unit circle. And so that area is going to be defined as 14th multiplied by pi, multiplied by the radius squared. And so this is going to simplify to be 1/4 multiplied by pi, multiplied by the radius squared, which is 3 squad, and this will further simplify to be 9 pi divided by 4. So the value of the second integral is going to be 9 pi divided by 4. Now if we were to take the two values and plug them back into the original statement, we will be left with negative 18 plus 9 pi divided by 4. And by distributing that -1, we'll be left with -18 minus 9 pi divided by 4, which is going to be the solution to the definite integral. And with that being said, the overall solution is going to be B. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Use geometry and properties of integrals to evaluate the following definite integrals.
∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍 . (Hint: Write the integral as sum of two integrals.)

Key Concepts

Definite Integrals

Properties of Integrals

Geometric Interpretation of Integrals

Watch next