Maths-
General
Easy

Question

The area bounded by y=3x and y equals x to the power of 2 end exponent is

  1. 10    
  2. 5    
  3. 4.5    
  4. 9    

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=3x and y equals x to the power of 2 end exponent. We have to find the area bounded by the given curve.

The correct answer is: 4.5


    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=3x and y equals x to the power of 2 end exponent.
    W e space c a n space s a y space t h a t colon
3 x space equals space x squared
x space equals space 3 space o r space 0
S o space n o w space t h a t space t h e space v a l u e space o f space x space i s space 0 space a n d space 3 space s o space t h e space l i m i t s space w i l l space b e space 0 space t o space 3. space
A r e a space equals integral subscript 0 superscript 3 left parenthesis 3 x space minus space x squared 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 fraction numerator 3 x squared over denominator 2 end fraction minus x cubed over 3 close square brackets subscript 0 superscript 3
A r e a space equals space open square brackets 3 cubed over 2 minus space 3 cubed over 3 minus 0 cubed over 2 plus space 0 cubed over 3 close square brackets
A r e a space equals space open square brackets 27 over 2 minus space 27 over 3 space close square brackets
A r e a space equals space open square brackets 27 over 2 minus space 9 space close square brackets
A r e a space equals space fraction numerator 27 minus 18 over denominator 2 end fraction
A r e a space equals space 9 over 2 equals 4.5

    So now here we can say that using the integration method, the area of the region bounded by the given curves is 4.5. 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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