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Current time:0:00Total duration:3:31

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

Video transcript

what I want to do in this video see if we can evaluate the definite integral from negative 3 to 3 of the square root of 9 minus x squared DX and I encourage you to pause this video and try it on your own and I'll give you a hint you can do this purely by looking at the graph of this function alright so I'm assuming you've had a go at it so let's just think about I just told you that you could do this using the graph of the function so let's graph this function so let's let's get a let's get a y-axis here so this is my y axis this is my x axis x axis and you might be saying oh well what is what is the graph of this thing it might not jump out at you it's been a little while it's as a hint since you've done conic sections maybe in your algebra class but let's just remind ourselves so if this this function so if we said Y is equal to some function of X which we see is the square root of 9 minus x squared then we could say well that means that Y squared must be equal to this thing squared which is 9 minus x squared and then we could say Y squared plus x squared is equal to 9 and you might recognize this as a circle circle centered at origin circle centered at origin with radius radius equal to 3 the square root of 9 so radius is equal to 3 centered at the origin now the graph of this is not going to be a circle this is a function this is a function so this is it would be a circle and it would not be a function anymore if you said the positive and negative square roots of year so when we took the square root we kind of got that or when we took the squares of both sides we got the the bottom back I guess you could say but up here we're only talking about the principal root so when you're talking about the principal root you're really talking about the top this is the top of a sort of a circle centered at the origin with radius 3 so this is top of circle because it's the positive square root so let's draw that it's gonna radius of three so centered at the origin so this is going to be negative three this is going to be three this is going to be three right over here so this graph this function is going to look like this it is going to look like this and it's actually only defined between negative 3 & 3 if you the absolute value of X is greater than 3 then you're gonna get a negative value in here and then you can't take the the principal root if we're defining it over over real if we're if we're defining it over positive or non-negative values I should say so this is the graph so what is the definite integral from negative 3 to 3 well it's just the area under the curve and above the x-axis is the stuff that I am shading in and green well what's that well you don't need calculus to figure that out you can do this with just traditional geometry the area of the entire the area of the entire circle if there were an entire circle will just be PI R squared so it would be pi times 3 squared which is equal to 9 pi now this is only half of the entire circle so we're going to divide that by 2 so the area is 9 PI over 2 so this thing is 9 PI over 2