Intuitive understanding of integration

Question

I am a high school student who has recently started calculus (all my previous calculus experience is mostly in physics calculating moment of inertia and other such stuff). The way I used to view the integral was as a sum of the values the function which I am integrating takes up over the whole domain with an infinitely small increment in the function's input value. But the problem I am facing is this- if I integrate sin x from 0 to pi/2, at least at two points it takes a value greater than 0.5. But the integration (as done in all standard places) comes to 1. What am I understanding wrong about what integration means itself?

Answer 1

I like to think of it like this: $\frac{1}{b-a} \int_a^b f(x)dx$ is the average value of $f$ on the interval $[a,b]$. I find it much easier and more intuitive to think about averages and the 'average value' of some function. Note this generalizes finite averages. So, $\int_a^b f(x)dx$ is the average value of $f$ times the length of the interval, which is the area under the curve of $f$.

So $\int_0^{\frac{\pi}{2}} \sin(x)dx = 1$ means that the average value of $\sin(x)$ on $[0,\frac{\pi}{2}]$ is $\frac{2}{\pi}$.

Answer 2

Integration gets the area underneath a curve.

You can have an very long line with an area of 0, right?

Answer 3

"What does integration mean?" is a really broad question. I'll give you an answer that might help give you some insight into the physics that you've learned.

"What does integration mean?". Answer: adding up "small" pieces.

You've seen this with the limit definition of the (Riemann) integral, adding up the areas of thin rectangles. You'll add up small "pieces" to get integral formulas for the volume and surface area of a solid of revolution and the length of a curve.

In physics you've probably saw this already when you calculate the moment of intertia of an object by splitting it into small chunks of mass $dm$ and then adding up the moment of inertia for each piece: $I = \displaystyle \int r^2 dm$.