Use: F(x)=0.2((x − 3)2)(x + 1), bounded by the x axis, over [-1,2] Use the limit definition of Riemman Sum to find the earea shaded beneath the curve, round to 3 decimal places.

Question

Use:  F(x)=0.2((x − 3)2)(x + 1), bounded by the x axis, over [-1,2]  Use the limit definition of Riemman Sum to find the earea shaded beneath the curve, round to 3 decimal places.

Brighton · Accepted Answer

Step 1  The length of the sub-intervals and the function is computed as follows,  Δ x=b − an  Δ x=3n  xi∗= −1 + 3in  Step 2  The formula for computing the area under the curve is,  ∫−12 0.2(x − 3)2(x + 1)dx=limn → ∞∑i=lnf(xi∗) Δ x  ∫−12(0.2x3 − x2 + 0.6x + 1.8)dx=limn → ∞ ∑i=lnf(−1 + 3in)(3n)  Step 3  Thus, the area of the function, using the limit definition of Riemann sum is,  limn → ∞ ∑i=lnf(−1 + 3in)(3n)=limn → ∞ ∑i=ln(0.2(−1 + 3in)3 − (−1 + 3in)2 + 0.6(−1 + 3in) + 1.8)(3n)  =limn → ∞ ∑i=ln(9.6in + 5.4i3n3 − 14.4i2n2)(3n)  =limn → ∞ ∑i=ln(28.8in2 + 16.2i3n4 − 43.2i2n3)  =0

Use: F(x)=0.2((x - 3)^{2})(x + 1), bounded by the x axis, over [-1,2] Use the limit definition of Riemman Sum to find the earea shaded beneath the curve, round to 3 decimal places.

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