Area under the curve MCQ Quiz - Objective Question with Answer for Area under the curve - Download Free PDF

Calculation

Given  &  & 

 is the image of  on 

Also,  passes through  & 

so bounded Area 

⇒ 48A = 10

Concept:

• The question involves finding the area bounded by inequalities.
• The inequalities form a region enclosed by y = 1/x, 5x − 4y − 1 = 0, 4x + 4y − 17 = 0, and the x-axis.
• To find the area, we:
  • Find the points of intersection of the given curves and lines.
  • Break down the area into simpler regions: triangles and integrals.
  • Use integration to find the area under the curve y = 1/x.
• Integration of 1/x: The integral of 1/x with respect to x is log_ex.
• The final area will be a combination of calculated triangular areas and definite integrals.

Calculation:

Given,

x > 0, y > 1/x, 5x − 4y − 1 > 0, 4x + 4y − 17

Points of intersection are calculated as follows:

⇒ 5x − 4y − 1 = 0 and y = 1/x meet at (1, 1)

⇒ 5x − 4y − 1 = 0 and 4x + 4y − 17 = 0 meet at (2, 1.25)

⇒ 4x + 4y − 17 = 0 and y = 1/x meet at (4, 0.25)

Break the area into:

Area of triangle with vertices (1,1), (2,1.25), (4,0.25)

Area of region under y = 1/x between x = 1 and x = 4

Area = (1/2) × (base 1.5) × (height 4/3)

⇒ 1/2 × 3/2 × 4/3 = 1

Next, area of another triangle:

Area = (1/2) × (base 2) × (height 10/4)

⇒ 1/2 × 2 × 2.5 = 2.5

Now subtract the area under the curve y = 1/x:

∫₁⁴ (1/x) dx = log_e4

Add all the areas:

Total Area = 1 + 2.5 − log_e4

Total Area = 33/8 − log_e4

∴ Hence, the area of the given region is 33/8 − log_e4.

So, the correct option is 2.

Calculation: 

Abscissa of the point of intersection of 2y = 5x² 

and y = x² + 6 is ± 2 

⇒ 

⇒ α³ = 600

Hence, the correct answer is 600. 

Concept:

Area of Region:

• The area between curves can be found by integrating the difference of the functions over the given interval.
• Use definite integrals and apply limits appropriately to find the enclosed area.

Calculation:

Given region:

Area calculation involves three integrals:

Evaluating each integral:

Substitute the values:

Given:

Comparing terms,

Hence, the correct answer is 22.

Concept:

The area under the curve y = f(x) between x = a and x = b,is given by,  Area = 

Calculation:

Here, we have to find the area of the region bounded by the curves y = , the line x = 2, x  = 0 and the x - axis

So, the area enclosed by the given curves = 

As we know that, 

Hence, option 4 is the correct answer.

Concept:

The area under the curve y = f(x) between x = a and x = b, is given by:

Area = 

Similarly, the area under the curve y = f(x) between y = a and y = b, is given by:

Area = 

Calculation:

Here, 

x² = y  and line y = 1 cut the parabola

∴ x² = 1

⇒ x = 1 and -1

Here, the area is symmetric about the y-axis, we can find the area on one side and then multiply it by 2, we will get the area,

This area is between y = x² and the positive x-axis.

To get the area of the shaded region, we have to subtract this area from the area of square i.e.

 square units.

Concept: 

Function y = √f(x) is defined for f(x) ≥ 0. Therefore y can not be negative.

Calculation:

Given: 

y =  and x-axis

At x-axis, y will be zero

y = 

⇒ 0 = 

⇒ 16 - x² = 0

⇒ x2 = 16

∴ x = ± 4

So, the intersection points are (4, 0) and (−4, 0)

Since the curve is y = 

So, y ≥ o [always]

So, we will take the circular part which is above the x-axis

Area of the curve, A 

We know that,

=  

= 

= 8 sin^-1 (1) + 8 sin-1 (1)

= 16 sin^-1 (1)

= 16 × π/2

= 8π sq units

Calculation:

Enclosed Area

Concept:

Area under a Curve by Integration

Find the area under this curve by summing vertically.

• In this case, we find the area is the sum of the rectangles, height x = f(y) and width dy.
• If we are given y = f(x), then we need to re-express this as x = f(y) and we need to sum from the bottom to top.

So, 

Calculation:

Given Curve: x = 4 - y2

⇒ y2 = 4 - x
⇒ y2 = - (x - 4)           

The above curve is the equation of the Parabola,

We know that at y-axis; x = 0

⇒ y2 = 4 - x

⇒ y2 = 4 - 0 = 4

⇒ y = ± 2

⇒ (0, 2) or (0, -2) are Point of intersection.

Area under the curve 

 Sq. unit

Given:

The parabolas y = 3x2 and x2 - y + 4 = 0

Concept:

Apply concept of area between two curves y₁ and y₂ between x = a and x = b

Calculation:

The parabolas y = 3x2 and x2 - y + 4 = 0

then 3x² = x² + 4

⇒ x² = 2

⇒ x = ± √ 2

Then the area is 

 sq unit.

Hence option (4) is correct.

Explanation:

Equation of circle is given by x² + y² = a²

Let's take the strip along a y-direction and integrate it from 0 to 'a' this will give the area of the first quadrant and in order to find out the area of a circle multiply by 4

Area of first Quadrant =  = 

Area of circle = 4 × 

Concept:

The area of the curve y = f(x) is given by:

A = 

where x1 and x2 are the endpoints between which the area is required.

Imp. Note: The net area will be the addition of the area below the x-axis and the area above the x-axis.

Calculation:

The f(x) = y = 4x³

Given the end points x1 = -2, x2 = 3

Area of the curve (A) =

⇒ A = 

⇒ A = 

⇒ A = 

⇒ A = 

⇒ A = 97

Additional Information

Integral property:

• ∫ xn dx = + C ; n ≠ -1
•  + C
• ∫ ex dx = ex+ C
• ∫ ax dx = (ax/ln a) + C ; a > 0,  a ≠ 1
• ∫ sin x dx = - cos x + C
• ∫ cos x dx = sin x + C 

Explanation:

Given equation of curves are

y = x²    ---(1) and y = 16    ---(2)

By solving both equation (1) and (2) we have:

x² = 16

x = 4, -4.

∴ Points of intersection are (4, 16) and (-4, 16).

From the figure we have,

By using Integral property we have,

Alternate Method 

There is another method also by which we can solved the problem,

By considering horizontal strip and by the condition of symmetry we have:

Area = 

Concept:

The area under a Curve by Integration:

Find the area under this curve is by summing horizontally.

In this case, we find the area is the sum of the rectangles, heights y = f(x) and width dx.

We need to sum from left to right.

∴ Area =  

Calculation: 

Here, we have to find the area of the region bounded by the curves y = x2, x-axis and ordinates x = -1 and x = 2

So, the area enclosed by the given curves is given by 

As we know that, 

Area = 

= 

= 

Area = 3 sq. units.

Concept:

The area under the function y = f(x) from x = a to x = b and the x-axis is given by the definite integral 

​

This is for curves that are entirely on the same side of the x-axis in the given range.

If the curves are on both sides of the x-axis, then we calculate the areas of both sides separately and add them.

Definite integral: If ∫ f(x) dx = g(x) + C, then 

.

Calculation:

.

Using the above concept for area of a curve, we can say that the required area is:

.