Sections in the Chapter - R D Sharma Objective Mathematics
1 Curve sketching - Hints
2 Sketching of some common curves
3 Areas of Bounded regions - Procedure for find areas
Study Plan
Day 1
1 Curve sketching - Hints
Day 2
2 Sketching of some common curves
3 Areas of Bounded regions - Procedure for find areas
Day 3
Revision of concepts
Obective Type Exercises 1 to 10
Day 4
O.T.E.: 11 to 30
Day 5
O.T.E.: 31 to 39
Practice Exercises 1 to 10
Day 6
Practice Exercises 11 to 30
Day 7
Practice Exercises 31 to 52
Revision Period
Concept revision
Test paper problem solving
Try to recollect relevant points on the topic.
If required right click on the topic and click on open in a new window to read the relevant material.
Close the window and come back.
Showing posts with label Area of bounded regions. Show all posts
Showing posts with label Area of bounded regions. Show all posts
Area of Bounded Regions
if f(x) is a continuous function defined on [a,b]. then, the area bounded by the curve y = f(x), the x-axis and the ordinates x = a and x = b is given by
∫f(x)dx(between limits a to b) or ∫y dx (between limits a to b).
Steps to find the area of bounded regions
1. visualize a rough sketch.
2. Slice the area into horizontal or vertical strips appropriately.
3. Create a formula for area treating the strip as a rectangle.
If the strip is parallel to y-axis, the width of the rectangle will be Δx.
If the strip is parallel to x-axis, the width of the rectangle will be Δy.
4. Find the limits of x (for vertical strips) and limits of y for horizontal strips.
5. Do the integration ∫ydx or ∫xdy
y = f(x) is the given function or x = f(y) is the given function.
∫f(x)dx(between limits a to b) or ∫y dx (between limits a to b).
Steps to find the area of bounded regions
1. visualize a rough sketch.
2. Slice the area into horizontal or vertical strips appropriately.
3. Create a formula for area treating the strip as a rectangle.
If the strip is parallel to y-axis, the width of the rectangle will be Δx.
If the strip is parallel to x-axis, the width of the rectangle will be Δy.
4. Find the limits of x (for vertical strips) and limits of y for horizontal strips.
5. Do the integration ∫ydx or ∫xdy
y = f(x) is the given function or x = f(y) is the given function.
Sketching of some common curves
Straight line
General equation = ax+by+c = 0
Circle
General equation = x² + y² + 2gx+2fy+c = 0
Standard parabola
y² = 4ax
General parabola
y = ax²+bx+c
Standard ellipse
x²/a² + y²/b² = 1
General equation = ax+by+c = 0
Circle
General equation = x² + y² + 2gx+2fy+c = 0
Standard parabola
y² = 4ax
General parabola
y = ax²+bx+c
Standard ellipse
x²/a² + y²/b² = 1
Curve Sketchig - Some Useful Hints
To evaluate the areas of bounded regions, one has to know the rough sketch of the curve.
The principles/ideas that help in visualizing the curve or rough sketching the curve are:
1. Origin and tangents at the origin
Check whether the curve passes through the origin and finding out whether (0,0) satisfies the curve equation.
If the curve passes through (0,0) find the equation of tangent at (0,0)
2. Points of intersection of the curve with the coordinate axes
You can find the point at which the curve crosses the x-axis by putting y = 0 in the curve equation.
Similarly putting x = 0 gives the point at which the curve passes y-axis.
3. Symmetry
If all powers of y in the curve equation are even, then the curve is symmetric about x-axis.
The familiar curve y² = 4ax is symmetric about x-axis.
If all powers x in the equation of curve are even, it is symmetric about y-axis.
If by putting x = -x and y = -y, the equation of the curve remains the same, then it symmetric in opposite directions.
If the equation remains unaltered by interchanging x and y in the equation, then it is symmetric about the line y =x.
4. Regions where the curve does not exist
In the regions where the curve does not exist, x or y values will be imaginary.
In the case of y² = 4ax, when x is negative, y values will be imaginary. Hence curve does not exist on the left side of y-axis.
5. Find points where dy/dx = 0
At these points, the slope of the tangent to the curve is zero and the tangent will be parallel to the x axis.
6. Find maxima and minima points
Find points where dy/dx = 0 and check the sign of d²y/dx² at these points and find out maximum (d²y/dx² is negative) and minimum (d²y/dx² is positive) points of the curve.
Also find the interval in which the dy/dx is g.t. 0. In this interval, the function is monotonically increasing.
In the interval in which dy/dx is l.t. 0, the function is monotonically decreasing.
The principles/ideas that help in visualizing the curve or rough sketching the curve are:
1. Origin and tangents at the origin
Check whether the curve passes through the origin and finding out whether (0,0) satisfies the curve equation.
If the curve passes through (0,0) find the equation of tangent at (0,0)
2. Points of intersection of the curve with the coordinate axes
You can find the point at which the curve crosses the x-axis by putting y = 0 in the curve equation.
Similarly putting x = 0 gives the point at which the curve passes y-axis.
3. Symmetry
If all powers of y in the curve equation are even, then the curve is symmetric about x-axis.
The familiar curve y² = 4ax is symmetric about x-axis.
If all powers x in the equation of curve are even, it is symmetric about y-axis.
If by putting x = -x and y = -y, the equation of the curve remains the same, then it symmetric in opposite directions.
If the equation remains unaltered by interchanging x and y in the equation, then it is symmetric about the line y =x.
4. Regions where the curve does not exist
In the regions where the curve does not exist, x or y values will be imaginary.
In the case of y² = 4ax, when x is negative, y values will be imaginary. Hence curve does not exist on the left side of y-axis.
5. Find points where dy/dx = 0
At these points, the slope of the tangent to the curve is zero and the tangent will be parallel to the x axis.
6. Find maxima and minima points
Find points where dy/dx = 0 and check the sign of d²y/dx² at these points and find out maximum (d²y/dx² is negative) and minimum (d²y/dx² is positive) points of the curve.
Also find the interval in which the dy/dx is g.t. 0. In this interval, the function is monotonically increasing.
In the interval in which dy/dx is l.t. 0, the function is monotonically decreasing.
Subscribe to:
Posts (Atom)