 # Ch08. Limit of Sum & Application of Definite Integrals

## Achievements

The following badges can be earned while learning.   ## Certificate

Certificate on successful completion of this course. In this online course, you will learn limit of sum, applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only), area between any of the two above said curves (the region should be clearly identifiable). For further understanding of concepts and for examination preparation, practice questions based on the above topics are discussed in the form of assignments that have questions from NCERT Textbook exercise, NCERT Examples, Board’s Question Bank, RD Sharma, NCERT Exemplar etc. instead of only one book. The PDF of assignments can be downloaded within the course.

## Course Content

Limit of Sum
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Area Under Curves
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The following list of questions are just meant for reference before purchasing membership. The list might or might not include NCERT Questions as it depends on the chapter/course. Some chapters have NCERT questions combined in the Assignments and some chapters have separate NCERT questions and Assignments. For complete details, please check the index of the course in the "About Course".

### Assignment – 1

Q1. {\displaystyle \int\limits_0^2 {({x^2} + 3)dx} = \frac{{26}}{3}}
Q2. {\displaystyle \int\limits_1^3 {({x^2} + x)dx = \frac{{38}}{3}} }
Q3. {\displaystyle \int\limits_0^2 {({x^2} + 2x + 1)dx = \,\frac{{26}}{3}} }
Q4. {\displaystyle \int\limits_0^4 {(x + {e^{2x}})dx = \frac{{15 + {e^8}}}{2}} }
Q5. {\displaystyle \int\limits_1^3 {(2{x^2} + 5x)\,dx = \,\frac{{112}}{3}} }
Q6. {\displaystyle \int\limits_1^3 {(2{x^2} + x + 9)\,dx} = \,\frac{{118}}{3}}
Q7. {\displaystyle \int\limits_2^3 {{e^{ - 4x}}dx} = \frac{{ - 1}}{4}({e^{ - 12}} - {e^{ - 8}})}
Q8. {\displaystyle \int_0^1 {{e^{2 - 3x}}} dx = \frac{1}{3}\left( {{e^2} - \frac{1}{e}} \right)}
Q9. {\displaystyle \int\limits_2^4 {{2^x}} dx = \frac{{12}}{{log2}}}
Q10. {\displaystyle \int_1^3 {({e^{2 - 3x}} + {x^2} + 1)\,dx} = \frac{1}{3}\left( {32 - \frac{1}{{{e^7}}} + \frac{1}{e}} \right)}

### Assignment – 2

Q1. Find the area enclosed by the circle {\displaystyle {x^2} + {y^2} = {a^2}} .
Q2. Find the area enclosed by the ellipse {\displaystyle \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1} .
Q3. Find the area of the region bounded by the curve {\displaystyle y = {x^2}} and the line {\displaystyle y = 4} .
Q4. Find the area of the region in the first quadrant enclosed by the x-axis, the line, the circle {\displaystyle {x^2} + {y^2} = 32} .
Q5. Find the area of region bounded by the curve {\displaystyle {y^2} = x} and the lines {\displaystyle x = 1,\,\,x = 4} , x-axis and the first quadrant.
Q6. Find the area of the region bounded by {\displaystyle {y^2} = 9x,\,\,x = 2,\,\,x = 4} and the x-axis in the first quadrant.
Q7. Find the area of the region bounded by {\displaystyle {x^2} = 4y,\,\,y = 2,\,\,y = 4} and the y-axis in the first quadrant.
Q8. Find the area of region bunded by the ellipse {\displaystyle \frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1} .
Q9. Find the area of region bounded by the ellipse {\displaystyle \frac{{{x^2}}}{4} + \frac{{{y^2}}}{9} = 1} .
Q10. Find the area of region in the first quadrant enclosed by x-axis, line {\displaystyle x = \sqrt 3 y} and the circle {\displaystyle {x^2} + {y^2} = 4} .
Q11. Find the area of the smaller part of the circle {\displaystyle {x^2} + {y^2} = {a^2}} cut off by the line {\displaystyle x = \frac{a}{{\sqrt 2 }}} .
Q12. The area between {\displaystyle x = {y^2}} and {\displaystyle x = 4} is divided into two equal parts by the line {\displaystyle x = a} , find the value of a.
Q13. Find the area bounded by the curve {\displaystyle {x^2} = 4y} and the line {\displaystyle x = 4y - 2} .
Q14. Find the area of the region bounded by the curve {\displaystyle {y^2} = 4x} and the line {\displaystyle x = 3} .
Q15. Find the area of the region bounded by the parabola {\displaystyle y = {x^2}} and {\displaystyle y = \,|x|} .

### Assignment – 3

Q1. Find the area lying in the first quadrant and bounded by the circle {\displaystyle {x^2} + {y^2} = 4} and the lines {\displaystyle x = 0} and {\displaystyle x = 2} .
Q2. Find the area of the region bounded by the curve {\displaystyle {y^2} = 4x} , y-axis and the line {\displaystyle y = 3} .
Q3. Find the area of the region bounded by the two parabolas {\displaystyle y = {x^2}} and {\displaystyle {y^2} = x} .
Q4. Find the area lying above x-axis and included between the circle {\displaystyle {x^2} + {y^2} = 8x} and the parabola {\displaystyle {y^2} = 4x} .
Q5. In figure, AOBA is the part of the ellipse {\displaystyle 9{x^2} + {y^2} = 36} in the first quadrant such that {\displaystyle OA = 2} and {\displaystyle OB = 6} . Find the area between the arc AB and the chord AB.

Q6. Using integration find the area of region bounded by the triangle whose vertices are (1, 0), (2, 2) and (3, 1).
Q7. Find the area of the region enclosed between the two circles: {\displaystyle {x^2} + {y^2} = 4} and {\displaystyle {(x - 2)^2} + {y^2} = 4} .
Q8. Find the area of the circle {\displaystyle 4{x^2} + 4{y^2} = 9} which is interior to the parabola {\displaystyle {x^2} = 4y} .
Q9. Find the area bounded by curves {\displaystyle {(x - 1)^2} + {y^2} = 1} and {\displaystyle {x^2} + {y^2} = 1} .
Q10. Find the area of the region bounded by the curves {\displaystyle y = {x^2} + 2} , {\displaystyle y = x} , {\displaystyle x = 0} and {\displaystyle x = 3} .
Q11. Using integration find the area of region bounded by the triangle whose vertices are (– 1, 0), (1, 3) and (3, 2).
Q12. Using integration find the area of the triangular region whose sides have the equations {\displaystyle y = 2x + 1,\,\,y = 3x + 1} and {\displaystyle x = 4} .

### Assignment – 4

Q1. Find the area of the parabola {\displaystyle {y^2} = 4ax} bounded by its latus rectum.
Q2. Find the area of the region bounded by the line {\displaystyle y = 3x + 2} , the x-axis and the ordinates {\displaystyle x = - 1} and {\displaystyle x = 1} .
Q3. Find the area bounded by the curve {\displaystyle y = \cos x} between {\displaystyle x = 0} and {\displaystyle x = 2\pi } .
Q4. Prove that the curves {\displaystyle {y^2} = 4x} and {\displaystyle {x^2} = 4y} divide the area of the square bounded by {\displaystyle x = 0} , {\displaystyle x = 4,\,\,y = 4} and {\displaystyle y = 0} into three equal parts.
Q5. Find the area of the region {\displaystyle \{ (x,\,\,y):0 \le y \le {x^2} + 1,\,\,0 \le y \le x + 1,\,\,0 \le x \le 2\,\} }
Q6. Find the area under given curves and given lines:
a. {\displaystyle y = {x^2},\,\,x = 1,\,\,x = 2} and x-axis.
b. {\displaystyle y = {x^4},\,\,x = 1,\,\,x = 5} and x-axis.
Q7. Find the area between the curves {\displaystyle y = x} and {\displaystyle y = {x^2}} .
Q8. Find the area of the region lying in the first quadrant and bounded by {\displaystyle y = 4{x^2},\,x = 0,\,y = 1} and {\displaystyle y = 4} .
Q9. Find the area bounded by the curve {\displaystyle y = \sin x} between {\displaystyle x = 0} and {\displaystyle x = 2\pi } .
Q10. Find the area enclosed between the parabola {\displaystyle {y^2} = 4ax} and the line {\displaystyle y = mx} .
Q11. Find the area enclosed by the parabola {\displaystyle 4y = 3{x^2}} and the line {\displaystyle 2y = 3x + 12} .
Q12. Find the area of smaller region bounded by the ellipse {\displaystyle \frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1} and the line {\displaystyle \frac{x}{3} + \frac{y}{2} = 1} .
Q13. Find the area of smaller region bounded by the ellipse {\displaystyle \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1} and the line {\displaystyle \frac{x}{a} + \frac{y}{b} = 1} .
Q14. Find the area of region enclosed by the parabola {\displaystyle {x^2} = y} , the line {\displaystyle y = x + 2} and the x-axis.
Q15. Using the method of integration find the area bounded by the curve {\displaystyle |x| + |y|\, = 1} .
Q16. Find the area bounded by the curve {\displaystyle y = {x^3}} , the x-axis and ordinates {\displaystyle x = - 2} and {\displaystyle x = 1} .
Q17. Find the area bounded by the curve {\displaystyle y = x|x|} , x-axis and the ordinates {\displaystyle x = - 1} and {\displaystyle x = 1} .
Q18. Find the area of circle {\displaystyle {x^2} + {y^2} = 16} exterior to the parabola {\displaystyle {y^2} = 6x} .
Q19. Find the area bounded by the y-axis, {\displaystyle y = \cos x} and {\displaystyle y = \sin x} when {\displaystyle 0 \le x \le \frac{x}{2}} .
Q20. Sketch the graph of {\displaystyle y = \,\,|x + 3|} and evaluate {\displaystyle \int_{ - 6}^0 {|x + 3|\,dx} } .
Q21. Find the area bounded by the ellipse {\displaystyle \frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1} and the ordinates {\displaystyle x = 0} and {\displaystyle x = ae} , where {\displaystyle {b^2} = {a^2}(\,1 - {e^2})} and {\displaystyle e < 1} .
Q22. Find the area of the region {\displaystyle \{ (x,y):\,{y^2} \le 4x,\,\,4{x^2} + 4{y^2} \le 9\} }
Q23. Find the area of the region {\displaystyle \{ (x,y):\,{x^2} + {y^2} \le 1 \le x + y\} }
Q24. Find the area bounded by semi-circle {\displaystyle y = \sqrt {25 - {x^2}} } and x-axis.
Q25. Using integration find the area of the region {\displaystyle \{ (x,\,y):{x^2} + {y^2} \le 2ax,\,\,{y^2} \ge ax,x,y \ge 0\} } .

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