9. Determine the value of ( int7^-1 f(x) d x ) given that (...

9. Determine the value of ( int_{7}^{-1} f(x) d x ) given that ( int_{13}^{7} f(x) d x=-9 ) and ( int_{13}^{-1} f(x) d x=-12 ). 10. Determine the value of ( int_{0}^{6} 4 f(x) d x ) given that ( int_{0}^{5} f(x) d x=10 ) and ( int_{5}^{6} f(x) d x=3 ). 11. Determine the value of ( int_{2}^{10} f(x) d x ) given that ( int_{2}^{4} f(x) d x=-1, int_{4}^{7} f(x) d x=3 ) and ( int_{10}^{7} f(x) d x=-8 ). 12. Determine the value of ( int_{-5}^{-1} f(x) d x ) given that ( int_{2}^{-5} f(x) d x=56, int_{7}^{2} f(x) d x=-90 ) and ( int_{-1}^{7} f(x) d x=45 ).

Transcript

00:01 To find the value of the integral from 7 to negative 1 of f of x dx, note that in rules of integrals, the lower limit has to be smaller than the upper limit, so we're going to flip this.

00:17 And in integration, if we have the integral from a to b of f of x dx, and we want to flip the limits, it's going to be equal to the negative of the flipped limits from b to a of f of x dx.

00:33 So the integral from 7 to negative 1 of f of x is going to equal the negative of the integral from negative 1 to 7 of f of x dx.

00:46 Now looking at their values in here, the boundaries are negative 1, 7, and 13.

00:53 So if we are to make a number line the smallest is negative 1 and then we are going to use 7 and then we have 13 so the integrals you can make out of these will be the integral from negative 1 to 13 of f of x dx will equal integral from negative 1 to 7 of f of x dx plus the integral from 7 to 13 of f of x dx and since what we want is the negative of the integral from negative 1 to 7 of f of x dx then we're going to transfer this to the left side and this to the right side it's going to give a negative of the integral from negative 1 to 7 f of x dx will equal the negative of the integral from negative 1 to 13 of f of x dx plus the integral from 7 to 13 of f of x dx looking at the values we have we have flipped limits so we're going to use their negatives to get the correct value for the integral then from here we should have the negative of the negative negative 12 so negative negative 12 plus the negative of negative 9 and that'll be negative 12 plus 9 equal to negative 3 for number 10 if you are to make the number line, you have values here which are 0, 5, and 6.

02:49 So to make the integral out of these, it'll be integral from 0 to 6 of f of x dx equal to the integral from 0 to 5 of f of x dx plus the integral from 5 to 6 of f of x dx.

03:10 The integral from 0 to 5 is 10 plus the integral from 5 to 6 is 3 so that gives us 13 but since we're looking for the integral from 0 to 6 of 4 f of x dx which is just 4 times the integral from 0 to 6 of f then this is just 4 times 13 which equals 52 for number 11 to find the integral from 2 to 10 of f of x dx we recall that the integral from 2 to 10 if the other boundaries are 2, 4, and 7.

03:50 So if this is 2, 4, 7, and that's 10, then it will be equal to the integral from 2 to 4 of f of x dx plus integral from 4 to 7 of f of x dx plus the integral from 7 to 10 of f of x dx so that'll be 2 to 4 is negative 1 plus 4 to 7 is 3 plus 10 to 7 that's the negative of 7 to 10 so that's negative of negative 8 we get negative 1 plus 3 plus 8 equal to 10.

04:41 And then lastly, to find the integral of negative 5 to negative 1 f of x dx, given these values here, we want to look at the smallest value used as a boundary and the other bounds in the integral.

04:56 We have negative 1, negative 5, 2 and 7.

05:01 So if we were to write that in a number line, this is negative 5, we have negative 1, and then we have 2, and then we have 7.

05:11 So if we were to write that, that's the integral from negative 5 to 7...

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