On the left, a man is pushing a crate up a straight incline. On the right, a man is pushing the same crate up a curving incline. The problem in both cases is to determine the amount of energy required to push the crate to the top. For the problem on the left, you can use algebra and trigonometry to solve the problem. For the problem on the right, you need calculus. Why do you need calculus with the problem on the right and not the left?
This is because with the straight incline, the man pushes with an unchanging force and the crate goes up the incline at an unchanging speed. With the curved incline on the right, things are constantly changing. Since the steepness of the incline is constantly changing, the amount of energy expended is also changing. This is why calculus is described as "the mathematics of change". Calculus takes regular rules of math and applies them to evolving problems.
With the curving incline problem, the algebra and trigonometry that you use is the same, the difference is that you have to break up the curving incline problem into smaller chunks and do each chunk separately. When zooming in on a small portion of the curving incline, it looks as if it is a straight line:
Then, because it is straight, you can solve the small chunk just like the straight incline problem. When all of the small chunks are solved, you can just add them up.
This is basically the way calculus works - it takes problems that cannot be done with regular math because things are constantly changing, zooms in on the changing curve until it becomes straight, and then it lets regular math finish off the problem. What makes calculus such a brilliant achievement is that it actually zooms in infinitely. In fact, everything you do in calculus involves infinity in one way or another, because if something is constantly changing, it is changing infinitely from each infinitesimal moment to the next. All of calculus relies on the fundamental principle that you can always use approximations of increasing accuracy to find the exact answer. Just like you can approximate a curve by a series of straight lines, you can also approximate a spherical solid by a series of cubes that fit inside the sphere.