Refer to the region R enclosed between the graph of the...

Refer to the region $R$ enclosed between the graph of the function $y=2 x-x^{2}$ and the $x$ -axis for $0 \leq x \leq 2$. (a) Sketch the region $R$. (b) Partition [0,2] into 4 subintervals and show the four rectangles that LRAM uses to approximate the area of $R$. Compute the LRAM sum without a calculator.

Transcript

00:01 Hi, in this question we are given with the function y that is equal to 2x minus x square.

00:08 For the first part, we need to sketch the region that is enclosed between the graph of the function y is equals to 2x minus x squared x x x is on an interval from 0 to and the graph of this region would look like this.

00:32 Equation y that is equals to 2x minus x square here this is our x -axis and we are talking about this region for the next part b we need to do the partition of region 0 to 2 into 4 sub interval and we need to show four rectangles that lram uses to approximate the area of this region for that the region would look like this and these are the four rectangles that we are talking about here in this case we will use interval from 0 to 2 into 4 sub intervals so we get our value for delta x which will be equals to 2 minus 0 by 4 that is equals to 1 by 2 so this is 0 this is 1 by 2 this is 1 and this is 3 by 2 and this is now we get our intervals as from 0 to 1 by 2 then from 1 by 2 to 1 then from 1 to 3 by 2 and then from 3 by 2 2 now the function value at 0 will be having a function 2 times 0 minus 0 square and this will be be equals to 0 then the function value at 1 by 2 it will be equals to 3 by 4 the function value at 1 will be 1 and the function value at 3 by 2 it will be equals to 3 by 4.

02:17 Now because we are using left remun sum lram and this will be equals to function value at 0 times delta x plus the function value at 1 by 2 times delta x plus the function value at 1 delta x plus the function value at 1 delta x plus the function value at 1 delta x plus the function value at 1 delta x plus the the function value at 3 by 2 delta x.

02:44 Here are delta x is equals to 1 by 2 and f0 is 0...

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