Estimate the area between y = sin(x^2) + 3 and the x-axis for 1 lessthanorequalt

Question

Estimate the area between y = sin(x^2) + 3 and the x-axis for 1 lessthanorequalto times lessthanorequalto 2 using the right-point rule and 10 rectangles 50 rectangles 100 rectangles Is this estimate approaching some limit as the number of rectangles is increased? Estimate the definite integral integral_0^2 Squareroot 1 + x^4 dx using the left-point rule and 10 rectangles 50 rectangles 100 rectangles Is this estimate approaching some limit as the number of rectangles is increased? Estimate the definite integral integral_-1^1 2^-x^2 dx using the midpoint rule and 10 rectangles 100 rectangles Is this estimate approaching some limit as the number of rectangles is increased?

Explanation / Answer

Question 1: interval is divided acccordingly , sum is taken and it isdivided by n in each case to get the area

Question 2: interval is divided acccordingly , sum is taken and it isdivided by n in each case to get the area

Question 3: interval is divided acccordingly , sum is taken and it isdivided by n in each case to get the area

x f(x) n=1 1.1 3.935616002 n=2 1.2 3.991458348 n=3 1.3 3.992903651 n=4 1.4 3.925211521 n=5 1.5 3.778073197 n=6 1.6 3.549355436 n=7 1.7 3.248946787 n=8 1.8 2.901751406 n=9 1.9 2.548534248 n=10 2 2.243197505 Sum 34.1150481 Area 3.41150481

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