Area Under Simple Curves

Step 1: Understand the Concept
To find the area of a region bounded by a curve and the y-axis, we use integration with respect to . If the curve is defined as , the area between the horizontal lines and is given by the definite integral:
Area =
In this case, the curve is , which is a parabola opening to the left.

Step 2: Analysis & Limits of Integration
First, we need to find the points where the curve intersects the y-axis (where the boundaries of our area lie). On the y-axis, the value of is always 0. Setting the equation to zero, we get:
.
Therefore, our limits of integration are from to .

Now, we set up the integral:
Area =
Integrating the terms individually:
Area =

Step 3: Calculation & Conclusion
We now apply the Fundamental Theorem of Calculus by substituting the upper limit and subtracting the substitution of the lower limit:
Upper limit ( ):
Lower limit ( ):

Subtracting them:
Area =
Area =
To simplify, convert 16 to a fraction: .
The area of the bounded region is square units.

Final Answer: (A)

Step 1: Concept
Area .
Step 2: Analysis
The curve intersects the y-axis where , so .
Area .
Step 3: Conclusion
.
Final Answer: (A)

Step 1: Concept
The latus rectum of is the line .
Step 2: Analysis
Area . .
Step 3: Conclusion
Area .
Final Answer: (A)

Step 1: Concept
Area .
Step 2: Analysis
Area .
Step 3: Conclusion
.
Final Answer: (B)

Step 1: Area
.
Step 2: Area
.
Step 3: Ratio
.
Final Answer: (d)

Step 1: Find the points of intersection.
We are given the curve and the line , which can be rewritten as:
Substitute this into the equation of the curve: Thus, the equation becomes: Solve this quadratic equation for : So, or .

Step 2: Find the corresponding -coordinates.
Substitute and into the equation :
- When , ,
- When , . So, the points of intersection are and .

Step 3: Set up the integral for the area.
The area between the curve and the line is given by the integral of the difference between the -values of the curve and the line:

Step 4: Solve the integral.
The integral can be solved using standard methods, but after performing the integration, the result does not match any of the given options.

Step 5: Conclusion.
Thus, the area bounded by the curve and the line does not correspond to any of the given options, so the correct answer is option (D).

Step 1: Set up the integral to find the area between the curves.
We need to calculate the area bounded by the curves , , and the line . We will find the area between these two curves over the interval from 0 to 1. The total area is given by:

Step 2: Evaluate the integral.
We need to compute: The integral of is , and the integral of is .

Step 3: Apply the limits of integration.
Now, substitute the limits from 0 to 1: This gives:

Step 4: Simplify the expression.
Simplifying the expression:

Step 5: Conclusion.
Thus, the area of the figure is , which corresponds to option (C).

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}



Step 3: Final Answer

Step 1: Formula / Definition}

Step 2: Calculation / Simplification}
The equation contains , which is not a polynomial in derivatives.
Degree is not defined.
Step 3: Final Answer

Concept: Use symmetry and integrate in first quadrant.

Step 1: Consider first quadrant.


Step 2: Find intersection.


Step 3: Set up area.


Step 4: Integrate.
Conclusion:
Area =

Concept: Find intersection points and integrate between curves.

Step 1: Find points of intersection.
Substitute into second:

Step 2: Express curves in terms of .


Step 3: Area between curves.


Step 4: Evaluate.

Concept: Graph is V-shaped with vertex at .

Step 1: Split interval.
From to :
From to :

Step 2: Compute area.


Step 3: Total area.

Concept: For modulus functions, split at the point where the expression inside modulus becomes zero.

Step 1: Split the interval

Step 2: First integral

Step 3: Second integral

Step 4: Total area Final: