Question #6243

Use inscribed rectangles to approximate the area under g(x) = -.05x^2 + 7 for -3 <= x <= 0 and rectangle width 0.5.

Expert's answer

Use inscribed rectangles to approximate the area under g(x) = -0.5x² + 7 for -3 <= x <= 0 and rectangle width 0.5.

First rectangle:

(x1 is the left base point, x2 is the right base point, S1 is rectangle's area)

x1 = -3; x2 = -2.5;

h(x1) = 2.5; h(x2) = 3.875;

S1 = min(h(x1),h(x2))*0.5 = 2.5*0.5 = 1.25;

Second rectangle:

x1 = -2; x2 = -1.5;

h(x1) = 3.875; h(x2) = 5.875;

S1 = 3.875*0.5 = 1.9375;

Third rectangle:

x1 = -1; x2 = -0.5;

h(x1) = 5.875; h(x2) = 6.875;

S1 = 5.875*0.5 = 2.9375;

Forth rectangle:

x1 = -0.5; x2 = 0;

h(x1) = 6.875; h(x2) = 7;

S1 = 6.875*0.5 = 3.4375;

So, approximated area is S = S1+S2+S3+S4 = 9.5625.

First rectangle:

(x1 is the left base point, x2 is the right base point, S1 is rectangle's area)

x1 = -3; x2 = -2.5;

h(x1) = 2.5; h(x2) = 3.875;

S1 = min(h(x1),h(x2))*0.5 = 2.5*0.5 = 1.25;

Second rectangle:

x1 = -2; x2 = -1.5;

h(x1) = 3.875; h(x2) = 5.875;

S1 = 3.875*0.5 = 1.9375;

Third rectangle:

x1 = -1; x2 = -0.5;

h(x1) = 5.875; h(x2) = 6.875;

S1 = 5.875*0.5 = 2.9375;

Forth rectangle:

x1 = -0.5; x2 = 0;

h(x1) = 6.875; h(x2) = 7;

S1 = 6.875*0.5 = 3.4375;

So, approximated area is S = S1+S2+S3+S4 = 9.5625.

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