Integration (Approximating area) 1) Write in sigma notation...

Integration (Approximating area) 1) Write in sigma notation and evaluate the sum of terms 2i for i = 3, 4, 5, 6. 2) Find the sum of the values of 4 + 3i for i = 1, 2,..., 100 3) Evaluate the sum indicated by the notation sum_{k=1}^{20}(2k + 1). 4) Sketch left-endpoint and right-endpoint approximations for f(x) = 1 / x on [1, 2]; use n = 4. Approximate the area using both methods. 5) a. Find an upper sum for f(x) = 10 - x^2 on [1, 2]; let n = 4. b. Sketch the approximation. 6) Using the function f(x) = sin x over the interval [0, pi / 2], find an upper sum; let n = 6.

Transcript

00:01 Hi, in this question we are given with different parts we need to solve them one by one starting from the first we need to write the sigma rotation for the given function and it would be written as sigma from i is equals to three to six to i.

00:16 Now if we solve for its sum it would be equals to two times three plus two times four plus two times five plus two times six and solving this for the weekend.

00:30 Get a final result that would be equals to 36 now moving on to the next part to we need to solve the submission that is given as for i is equals to one 200 four plus three i as this could be further written as four times submission i is equals to one two hundred one plus three times summation i is equals to one to hundred i as we know this would be further equal to four times hundred because sum of all these would result in hundred plus three times here we can use the formula for i numbers to be added which would be equals to hundred times one plus hundred by two and solving this further we will get a final result and the summation for this would be equals to 15 ,550.

01:30 Now moving on to the next part third where we need to evaluate the given submission which is given as for k is equals to 1 to 20 2k plus 1 as this could be further written as equals to 2 times submission from k is equals to 1 to 20 plus submission from k is equals to 1 to 21 and solving this further we get this to be equals to 2 times 20 into 20 plus 1 by 2 plus 20.

02:06 And solving this further, we get a final result which would be equals to 440.

02:12 Now moving on to the next part, fourth one, where we need to go for the sketching of end point for approximation given as the function fx is equals to 1 by x.

02:24 For this, first we calculate delta x which would be equals to 2 minus 1 by 4 that is equals to 1 by 4 so in this case our intervals would be like for 1 .25 1 .25 1 .5 1 .5 and 1 .75 then 1 .75 and 2 .5 now if we go for the lower sum it would be equals to delta x that is 1 by 4 times f of 1 plus f of 1 .25 plus f of 1 .5 plus f of 1 .75...

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