Area: Find the area between the x-axis and the curve y = √(1 +...

Question

Area: Find the area between the x-axis and the curve y = √(1 + cos 4x), for 0 ≤ x ≤ π.

Accepted Answer

Welcome back everyone. Find the area of the region bounded by the curve y equals square root of 1 plus sin 2 x, d x axis, and the vertical lines x equals 0 and x equals pi divided by 2 A 4 square units, b 6 square units, c 2 square units, and d 1 square unit. So for this problem, let's remember how we identify. The area bounded by the curve and the x axis we have area equals integral from a to b of the absolute value of fx d x. In this case, notice that our curve corresponds to the positive square root of a function. So basically we are not expected to use the absolute value. In particular, we can rewrite the integral as integral from 0 to pi divided by 2. Those are the bounds given in the problem, and we want to integrate the square root of 1 plus sin 2 x dx. Now how do we integrate? Well, essentially we can remember one of the identities. 1 plus sin 2 x and trigonometry can be written as sin x plus cosine x squared. And now if we take square root, well, we're going to have The absolute value of sin x plus cosine x and we're choosing the positive solution, right? Additionally, sin x plus cosine x is non-negative on the given interval, so we can now show that the area equals integral from 0 to pi divided by 2 of. Sin x plus cosine x dx. Well done and now we can integrate the integral of sine is going to be equal to negative cosine, so we get negative cosine x, and the integral of cosine is sine, so we get sin x, and we want to evaluate. The result from 0 to pi divided by 2. Applying the fundamental theorem of calculus part 2, we get negative cosine of pi divided by 2 plus sine of pi divided by 2 minus negative cosine of 0 plus sine of 0. Now cosine of pi divided by 2 is 0. sin of 0 is 0, and for the remaining terms we get 1 for sin of pi divided by 2 minus negative cosine of 0. So we're basically adding cosine of 0, which is 1. And basically the final answer becomes 2 square units, which is option C. Thank you for watching.

Area: Find the area between the x-axis and the curve y = √(1 + cos 4x), for 0 ≤ x ≤ π.

Watch next