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General
Easy

Question

The area bounded by y equals x squared plus 2 comma X- axis, x=1 and x=2 is

  1. fraction numerator 16 over denominator 3 end fraction    
  2. fraction numerator 17 over denominator 3 end fraction    
  3. fraction numerator 13 over denominator 3 end fraction    
  4. fraction numerator 20 over denominator 3 end fraction    

Hint:

Integration, as we all know, is the process of determining an area by first dividing it into a number of basic strips and then adding up each one. At this point, we can calculate the area enclosed by a curve and a line connecting a given set of points. Here we have given the curve y equals x squared plus 2 comma X- axis, x=1 and x=2. We have to find the area bounded by the given lines and curve.

The correct answer is: fraction numerator 13 over denominator 3 end fraction


    We are aware that in a planar lamina, the region inhabited by two-dimensional forms is expressed as an area. Calculus requires that you know the difference between two definite integrals of a function in order to calculate the area between two curves. The definite integral of one function, such as f(x), minus the definite integral of other functions, such as g(x), with the lower and upper bounds as a and b, respectively, is used to define the area between the two curves or functions.
    Here we have given the curve as y equals x squared plus 2 comma X- axis, x=1 and x=2. 
    solution

    Refering this image, we can say that the integration will be:
    A r e a space equals integral subscript 1 superscript 2 y d x
A r e a space equals integral subscript 1 superscript 2 left parenthesis x squared plus 2 right parenthesis d x
I n t e g r a t i n g space i t space m a n u a l l y comma space w e space g e t colon
A r e a space equals open square brackets x cubed over 3 plus 2 x close square brackets subscript 1 superscript 2
A r e a space equals space open square brackets 2 cubed over 3 plus 2 left parenthesis 2 right parenthesis space minus space 1 cubed over 3 minus 2 left parenthesis 1 right parenthesis close square brackets
A r e a space equals space open square brackets 8 over 3 plus 4 space minus space 1 third minus 2 close square brackets
A r e a space equals space 7 over 3 plus 2
A r e a space equals space 13 over 3

    So now here we can say that using the integration method, the area of the region bounded by the given curve and the lines is 13/3. The equation A = ∫ab f(x) dx gives the area under the curve y = f(x) and x-axis. The bounding values for the curve with respect to the x-axis are shown here as a and b.

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