Find the area enclosed by the polar curve r=4e^(0.8θ) on the interval 0≤θ≤17 and the straight line segment between its ends.

Question

I found the area from the region to the polar curve, it's 5(e^(8/35)-1), however this is not the full answer. I need to subtract the area of the triangle formed by the x axis the segment between the ends of the curve and the segment connecting the origin with the endpoint of the curve. However, when I graph this equation its a spiral. I don't know how to do this second part of the problem.

Answer 1

This my my take, which might be wrong.

First I assume that θ is measured in degrees, not radians; as you said, the curve is a spiral

and the question would not make much sense if θ were in radians.

Secondly the value I got for the region to the polar curve is 5(e^27.2-1), and I do not

know how to account for the difference with your result

Your question is about the triangle.

The x-axis segment of the triangle is 4.

The segment connecting the origin with the end point is 4e^0.8x17 = 4e^13.6.

The height of the triangle is then 4e^13.6sin(17°).

So the area of the triangle is 8e^13.6sin(17°).