Area / functionsCalculate the area between f(x)=3x^2+2x and x=1 and x=2.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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To calculate the required area between f(x)=3x^2+2x and x=1 and x=2, we need the definite integral of f(x) between x = 1 and x = 2

Int[ 3x^2 + 2x dx], x =1 to x = 2

=> 3x^3/3 + 2x^2/2 , x = 1 to x = 2

=> x^3 + x^2, x = 1 to x = 2

=> 2^3 + 2^2 - 1^3 - 1^2

=> 8 + 4 - 1 - 1

=> 10

The required area is 10 square units.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

Since there is not established the other boundary curve, we'll suppose that we have to calculate the area between f(x), the lines x=1 and x=2 and the x axis.

The definite integral will be calculated with Leibniz-Newton formula:

Int f(x)dx = F(b)-F(a)

We'll calculate the indefinite integral of f(x):

Int f(x)dx = Int (3x^2 + 2x)dx

We'll use the property of integral to be additive:

Int (3x^2 + 2x)dx = Int 3x^2dx + Int 2xdx

Int 3x^2dx = 3*x^3/3 + C

Int 3x^2dx =x^3 + C

Int 2xdx = 2*x^2/2 + C

Int 2xdx = x^2 + C

Int (3x^2 + 2x)dx = x^3 + x^2 + C

F(2) - F(1) = 2^3 + 2^2 - 1^3 - 1^2

F(2) - F(1) = 8 + 4 - 2

F(2) - F(1) = 10

The area bounded by the curve of f(x) and the lines x=1, x=2 and x axis is A=10.

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