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1. ⇒  (MHT CET 2023 11th May Evening Shift )

The slope of the tangent to a curve y = f ( x ) at ( x , f ( x ) ) is 2 x + 1 . If the curve passes through the point ( 1 , 2 ) , then the area (in sq. units), bounded by the curve, the X -axis and the line x = 1 , is

A. 3 2

B. 4 3

C. 5 6

D. 1 12

Correct Option is (C)

MHT CET 2023 11th May Evening Shift Mathematics - Area Under The Curves Question 3 English Explanation

According to the given condition, d y   d x = 2 x + 1

Integrating w.r.t. x , we get y = x 2 + x + c

As curve passes through ( 1 , 2 ) , we get 2 = ( 1 ) 2 + 1 + c c = 0

The equation of the curve is y = x 2 + x .

Required area

= 0 1 ( x 2 + x ) d x = [ x 3 3 + x 2 2 ] 0 1 = 1 3 + 1 2 = 5 6  sq. units 

2. ⇒  (MHT CET 2023 10th May Evening Shift )

The area bounded by the X -axis and the curve y = x ( x 2 ) ( x + 1 ) is

A. 37 12 sq. units

B. 27 12 sq. units

C. 37 4 sq. units

D. 27 13 sq. units

Correct Option is (A)

For X-axis,  y = 0 x ( x 2 ) ( x + 1 ) = 0 x = 0  or  x = 2  or  x = 1

MHT CET 2023 10th May Evening Shift Mathematics - Area Under The Curves Question 5 English Explanation

Required area = 1 0 y d x + | 0 2 y d x |

= 1 0 x ( x 2 ) ( x + 1 ) d x + | 0 2 x ( x 2 ) ( x + 1 ) d x | = 1 0 ( x 3 x 2 2 x ) d x + | 0 2 ( x 3 x 2 2 x ) d x | = [ x 4 4 x 3 3 x 2 ] 1 0 + | [ x 4 4 x 3 3 x 2 ] 0 2 | = 5 12 + | 8 3 | = 37 12  sq. units 

3. ⇒  (MHT CET 2023 10th May Morning Shift )

Area of the region bounded by the curve y = 49 x 2 and X -axis is

A. 49 π sq. units

B. 49 π 2 sq. units

C. 49 π 4 sq. units

D. 98 π sq. units

Correct Option is (B)

MHT CET 2023 10th May Morning Shift Mathematics - Area Under The Curves Question 6 English Explanation

 Required area  = 2 0 7 49 x 2   d x = 2 [ x 2 49 x 2 + 49 2 sin 1 ( x 7 ) ] 0 7 = 2 [ 7 2 49 49 + 49 2 sin 1 ( 7 7 ) ] 0 = 2 × 49 2 × π 2 = 49 π 2  sq. units 

4. ⇒  (MHT CET 2023 9th May Evening Shift )

The area (in sq. units) bounded by the curve y = x | x | , X -axis and the lines x = 1 and x = 1 is

A. 2 3

B. 1 3

C. 1

D. 4 3

Correct Option is (A)

y = x | x | [  Given  ]  Required area  = 1 1 x | x | d x = 2 0 1 x 2   d x ... [ Area is always positive] = 2 × [ x 3 3 ] 0 1 = 2 × ( 1 3 0 ) = 2 3  sq.units

5. ⇒  (MHT CET 2021 21th September Evening Shift )

The area bounded by the parabola y 2 = x , the straight line y = 4 and Y axis is

A. 2 7 sq. units

B. 64 3 sq. units

C. 16 3 sq. units

D. 7 2 sq. units

Correct Option is (B)

MHT CET 2021 21th September Evening Shift Mathematics - Area Under The Curves Question 10 English Explanation

Refer figure

Required area is shaded.

Point of intersection of y 2 = x and y = 4 is A ( 16 , 4 ) .

A = 0 4 y 2 d y = [ y 3 3 ] 0 4 = 64 3  sq. units