Question

Can somebody solve this problem?

Answer 1

Alright, so first we need to dissect this sum:
In order to find the area, we need to approximate the function with rectangles. The pi/8n is the width of each slim rectangle, and the tan(i*pi/8n) is the height of each rectangle. The limit essentially says that if we make infinite, infinitely thin rectangles (the width will approach 0), our approximation becomes an equality.

In the problem they ask you for the interval. imagine n is some really big number, like 1,000,000. When i is at its lowest value, you can see that tan's argument will be essentially 0, so that is our left interval (you got that already). However, as i gets larger and approaches n, tan's argument approaches pi/8, so that will be our rightmost bound.

i/n will always be a number between 0 and 1, so in that way it can be thought of our "progress". by multiplying our progress by the actual length of our interval, pi/8, we can get our current position on the x axis. the only place where we use this value for x (i*pi/8n) is as the input to our tan function. Thus, our f(x) is tan(x)

Let me know if you still have any problems!