Determine whether the following statements are true and give a...

Question

Determine whether the following statements are true and give an explanation or counterexample.

c. ∫₀¹(x−x^2) dx=∫₀¹(√y−y) dy

Accepted Answer

Welcome back, everyone. Is the following Equality true? Integral from 0 to 1 of X2 minus X cub to DX equals integral from 0 to 1 of Y minus Y raises to the power of 3 halves Dy. We have to answer choices A's as true and B's as false. What we're going to do in this problem is simply integrate each expression, integral from 0 to 1 of X squad minus X cubed DX. Can be found using the power rule where integrating X2 is integral is X cubed divided by 3. And now we can apply the power rule for the second part, so we're subtracting the integral of X cubed, which is X to the power of 4 divided by 4, and we want to evaluate the result from 0 to 1 using the fundamental theorem of calculus part 2. That would be 1 cubed divided by 3 minus 1, raise to the power of 4 divided by 4, and then we're subtracting the result at 0. So that would be 0 cubed divided by 3 minus 0 to the power of 4 divided by 4. This is equal to 1/3 minus 1/4. We can find the common denominator which is 12 and then subtract 4 minus 3, which is 1. So, the difference is 1/12, and this is the value of the integral, right? Now let's go ahead and evaluate the integral from 0 to 1 of Y minus Y to the power of 3 halves D Y. Once again we can apply the power rule. The integral of Y is Y2d divided by 2. And the integral of Y raises the power of three halves is Y raises the power of 5 halves divided by 5 halves. And we want to evaluate the result from 0 to 1. Let's simplify this is Y2 divided by 2 minus 25. Why raise the power of 5 halves? We are evaluating the result from 0 to 1. We get 1 squared divided by 2 minus 25. Multiplied by 1 raise the power of 5 halves and we are subtracting the result at 0, so 0 square divided by 2 minus 2/5 multiplied by 0 raises the power of 5 halves. This is equal to 1/2. Minus 2/5. And then we're subtracting 0, right? Once again, we can find the common denominator, which is 10 to multiplied by 5. And in the numerator we are subtracting 5 minus. 2 multiplied by 2, that's 4, so 5 minus 4 is 1, and we get a 1/10. And then we can compare the results because one's wealth is not equal to 17. Since 12 is not equal to 10, the answer is be false. These integrals are not equal to each other. Thank you for watching.

Determine whether the following statements are true and give an explanation or counterexample.c. ∫₀¹(x−x^2) dx=∫₀¹(√y−y) dy

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Determine whether the following statements are true and give an explanation or counterexample.

c. ∫₀¹(x−x^2) dx=∫₀¹(√y−y) dy