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### Course: Class 11 Physics (India)>Unit 5

Lesson 3: Definite integral as area

# Worked example: Definite integral by thinking about the function's graph

If you already know the area under a curve, you can use it to compute an integral.

## Want to join the conversation?

• When we solve definite integral, do we always have to graph the function?
Also, why does definite integral look like the Anti-derivative symbol?
• No, we often solve definite integrals without graphing them, although graphing is often helpful in understanding the problem and checking to see if our result is reasonable. As for the symbols, it may seem odd that we use the same one for two entirely different concepts, one of which yields a function while the other yields a constant -- but the relationship between these two concepts is so important that its proof is known as the Fundamental Theorem of Calculus. It is our ability to use antiderivatives to solve definite integrals that gives calculus much of its power.
• where are the intro videos of definite integral he started this from simply evaluating it
• how does he know that it just includes the top part of the circle?
• The equation of a circle centred at the origin with radius 3 is x² + y² = 9. Hence y² = 9 - x². But then we have y = √(9 - x²), or we have y = -√(9 - x²). Since he is dealing with √(9 - x²) in the video, we know that he is dealing with points on a circle with y ≥ 0. Hence only the "top" of the circle.
• Why is the radius the square root of x^2 + y^2?
• The Pythagorean theorem states that x² + y² = r². Now take the square root of this.

On a deeper level, that is the way we compute distance in R², using the Euclidean norm, or the Euclidean metric, but you do not have to worry about such things as a normed linear space or a metric space at the moment. Go with the short answer at the top...
• so integral means finding the area below the graph ?
• That is one use for integrals. There are other uses.
• Hey I just wonder why we use the integral notation in indefinite integral rather than directly use f(x)dx ? It makes sense when we use the "s"notation from a to b in definite integral which I regard it's meaning is same as the "sum" of f(x)dx when there is a infinite point from a to b
• This is a keen observation. We have the elongated "s" without the upper and lower bounds ( b and a respectively) for indefinite integrals because they are actually related to definite integrals. The indefinite integral is defined as the area under a function f(u) from u=a ( a being a constant) to u=x (x being a variable). The definite integral is a function of the upper and lower bounds, you can interpret the indefinite integral as being a function of just the upper bound, ( we are holding the lower bound a as constant and making the upper bound b vary which is denoted by changing b to x)

We omit the upper bound u=x and the lower bound u=a to keep things simple.
• If we HAD been taking the definite integral of +/- sqrt(9-x^2), and the graph had just been a circle, would it have summed to zero?