{"id":14664,"date":"2024-03-28T12:54:53","date_gmt":"2024-03-28T07:24:53","guid":{"rendered":"https:\/\/tsboardsolutions.in\/?p=14664"},"modified":"2024-04-01T17:42:05","modified_gmt":"2024-04-01T12:12:05","slug":"ts-inter-2nd-year-maths-2b-definite-integrals-important-questions","status":"publish","type":"post","link":"https:\/\/tsboardsolutions.in\/ts-inter-2nd-year-maths-2b-definite-integrals-important-questions\/","title":{"rendered":"TS Inter 2nd Year Maths 2B Definite Integrals Important Questions"},"content":{"rendered":"

Students must practice these\u00a0TS Inter 2nd Year Maths 2B Important Questions<\/a> Chapter 7 Definite Integrals to help strengthen their preparations for exams.<\/p>\n

TS Inter 2nd Year Maths 2B Definite Integrals Important Questions<\/h2>\n

Very Short Answer Type Questions<\/span><\/p>\n

Question 1.
\nEvaluate \\(\\int_1^2 x^5\\) dx
\nSolution:
\n\\(\\int_1^2 x^5 d x=\\left[\\frac{x^6}{6}\\right]_1^2=\\frac{2^6}{6}-\\frac{1}{6}=\\frac{64}{6}-\\frac{1}{6}=\\frac{63}{6}=\\frac{21}{2}\\)<\/p>\n

Question 2.
\nEvaluate \\(\\int_0^\\pi \\) sinx dx
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 3.
\nEvaluate \\(\\int_0^a \\frac{d x}{x^2+a^2}\\)
\nSolution:
\n\"TS<\/p>\n

Question 4.
\nEvaluate \\(\\int_1^4 x \\sqrt{x^2-1}\\) dx
\nSolution:
\nLet x2<\/sup> – 1 – t \u21d2 2x dx dt then
\nUpper limit when x = 4 is t = 15.
\nLower Limit when x = 1 is t = 0.
\n\"TS<\/p>\n

Question 5.
\nEvaluate \\(\\int_0^2 \\sqrt{4-x^2}\\) dx
\nSolution:
\nLet x= 2 sin \u03b8 = dx – 2cos\u03b8 d\u03b8\u00a0then
\n\"TS<\/p>\n

Question 6.
\nShow that \\(\\int_0^{\\frac{\\pi}{2}} \\sin ^n x d x=\\int_0^{\\frac{\\pi}{2}} \\cos ^n x dx\\)
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 7.
\nEvaluate \\(\\int_0^{\\frac{\\pi}{2}} \\frac{\\cos ^{\\frac{5}{2}} x}{\\sin ^{\\frac{5}{2}} x+\\cos ^{\\frac{5}{2}} x}\\) dx
\nSolution:
\n\"TS<\/p>\n

Question 8.
\nEvaluate \\(\\int_0^{\\frac{\\pi}{2}} \\) x sin x dx
\nSolution:
\n\"TS<\/p>\n

Question 9.
\nEvaluate
\n\"TS
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

(iii) \\(\\int_0^{\\frac{\\pi}{2}} \\sin ^6 x \\cos ^4 x dx\\)
\nSolution:
\n\"TS<\/p>\n

Question 10.
\nFind \\(\\int_{-\\frac{\\pi}{2}}^{\\frac{\\pi}{2}} \\sin ^2 x \\cos ^4 x d x\\)
\nSolution:
\n\"TS<\/p>\n

Short Answer Type Questions<\/span><\/p>\n

Question 1.
\nFind \\(\\int_0^2\\left(x^2+1\\right) dx\\) as the limit of a sum
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 2.
\nEvaluate \\(\\int_0^2 e^x dx\\) as the limit of a sum.
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 3.
\nLets define f : [0,1]\u2192 R by
\nf(x) = 1 if x is rational
\n= 0 if x is irrational
\nthen show that f is nor R Integrable over [0, 1].
\nSolution:
\nLet P = (x0<\/sub>, x1<\/sub>,…., xn<\/sub>] be a partition of [0, 1].
\nSince between any two real numbers there exists rational and irrational numbers and
\nlet ti<\/sub>, si<\/sub> \u2208 [Xi<\/sub> -i xj<\/sub>] be the rational and irrational numbers.
\n\"TS<\/p>\n

Question 4.
\nEvalute \\(\\int_0^{16} \\frac{x^{\\frac{1}{4}}}{1+x^{\\frac{1}{2}}}\\) dx
\nSolution:
\nLet x = t4<\/sup> then dx – 4t3<\/sup> dt
\nUpper limit when x = 16 is t = 2.
\nand Lower limit when x = 0 is t = 0.
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 5.
\nEvaluate \\(\\int_{-\\frac{\\pi}{2}}^\\pi \\sin\\) |x| dx
\nSolution:
\nWe have sin |x| = sin(-x) if x < 0
\n= sinx if x \u2265 0
\n\"TS<\/p>\n

Question 6.
\nEvaluate by using the method of finding definite integral as the limit of a sum.
\n\\(\\lim _{n \\rightarrow \\infty} \\sum_{i=1}^n \\frac{1}{n}\\left(\\frac{n-1}{n+1}\\right)\\)
\nSolution:
\n\"TS<\/p>\n

Question 7.
\nEvaluate \\(\\lim _{n \\rightarrow \\infty} \\frac{2^k+4^k+6^k+\\ldots+(2 n)^k}{n^{k+1}}\\) using the method of finding definite integral\u00a0as the limit of a sum.
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 8.
\nEvaluate \\(\\lim _{n \\rightarrow \\infty}\\left[\\left(1+\\frac{1}{n}\\right)\\left(1+\\frac{2}{n}\\right) \\ldots\\left(1+\\frac{n}{n}\\right)\\right]^{\\frac{1}{n}}\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 9.
\nObtain Reduction formula for \\(\\int_0^{\\frac{\\pi}{2}} \\sin ^n x d x\\) and hence find
\n\"TS
\nSolution:
\n\"TS
\n\"TS
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 10.
\nEvaluate \\(\\int_0^a \\sqrt{a^2-x^2} dx\\)
\nSolution:
\nLet x = a sin\u03b8 then dx = a cos\u03b8 d\u03b8
\nUpper limit when x = a is \u03b8 = \\(\\frac{\\pi}{2}\\)
\nand Lower limit when x = 0 is \u03b8 = 0
\n\"TS<\/p>\n

Question 11.
\nFind \\(\\int_{-a}^a x^2\\left(a^2-x^2\\right)^{3 \/ 2} dx\\)
\nSolution:
\nSince f(x) = x2<\/sup>\u00a0(a2<\/sup> – x2<\/sup>)3\/2 <\/sup>is an even function and f(- x) = f(x) we have
\n\\(\\int_{-a}^a x^2\\left(a^2-x^2\\right)^{3 \/ 2} d x=2 \\int_0^a x^2\\left(a^2-x^2\\right)^{3 \/ 2} d x\\)
\nLet x = a sin \u03b8 then dx = a cos \u03b8 d\u03b8
\n\u2234 Upper limit when x = a is \u03b8 = \\(\\frac{\\pi}{2}\\)
\nLower limit when x = 0 is \u03b8 = 0
\n\"TS<\/p>\n

Question 12.
\nFind \\(\\int_0^1 x^{3 \/ 2} \\sqrt{1-x} dx\\)
\nSolution:
\nLet x = sin2<\/sup>\u03b8 then dx = 2 sin\u03b8 cos\u03b8 d\u03b8
\nUpper limit when x = 1 is \u03b8 = \\(\\frac{\\pi}{2}\\)
\nLower Limit when x = \u03b8 is \u03b8 = 0.
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 13.
\nFind the area under the curve f(x) = sin x in (0, 2\u03c0).
\nSolution:
\nConsider the graph of the function f(x) = sinx in [0, 2\u03c0];
\nwe have sin x \u2265 0 \u2200 x \u2208 [0,\u03c0] and sin x\u22640\u2200x\u2208[\u03c0,2\u03c0].
\n\"TS<\/p>\n

Question 14.
\nFind the area under the curve f(x) = cos x in [0, 2\u03c0].
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 15.
\nFind the bounded by the y = x2<\/sup> parabola the X- axis and the lines x = – 1, x = 2.
\nSolution:
\n\"TS<\/p>\n

Question 16.
\nFind the area cut off between the line y = 0 and the parabola y = x2<\/sup>– 4x + 3.
\nSolution:
\nThe point of intersection of y – 0 and y = x2<\/sup> – 4x + 3 is given by x2<\/sup> – 4x + 3 = 0
\n= (x – 3)(x-1) = 0 = x = 1 or 3
\ny=x2<\/sup>– 4x + 3 \u21d2 y+1 =\u00a0 x2<\/sup>– 4x + 4 (x-2)2<\/sup>
\nHence the equation represents a parabola
\nwith vertex (2, -1) lies in IV quadrant.
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 17.
\nFind the area bounded by the curves y = sin x and y = cos x between any two consecutive points of intersection.
\nSolution:
\nThe given curves y = sin x and y = cosx and
\ntan x = 1 \u21d2 x = \\(\\frac{\\pi}{4}\\)
\n\u2234 x = \\(\\frac{\\pi}{4}\\) and x = \\(\\frac{5 \\pi}{4}\\) are the two consecutive points of intersection.
\nTaking f(x) = sin x and g(x) cos x over \\(\\left[\\frac{\\pi}{4}, \\frac{5 \\pi}{4}\\right]\\) we have
\nf(x)> g(x) \u2200 x \u2208\\(\\left[\\frac{\\pi}{4}, \\frac{5 \\pi}{4}\\right]\\).
\nHence the area bounded by y = sin x, y = cos x and the two points of intersection is
\n\"TS<\/p>\n

Question 18.
\nFind the area of one of the curvilinear rectangles bounded by y = sin x, y cos x and X-axis.
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 19.
\nFind the area of the right angled triangle with base b and altitude \u2018h\u2019 using the fundamental theorem of integral calculus.
\nSolution:
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 20.
\nFind the area bounded between the curves y2<\/sup> – 1 = 2x and x = 0.
\nSolution:
\nThe given curves are
\ny2<\/sup>-1-2x-2(x-0) ……………. (1)
\n= (y-0)2 <\/sup>2(x)+1=2 \\(\\left[\\mathrm{x}+\\frac{1}{2}\\right]\\)
\n(1) represents parabola with vertex \\(\\left(-\\frac{1}{2}, 0\\right)\\)
\nSolving (1) and x = 0 we get
\ny2<\/sup> -1 = 0 \u21d2 y = \u00b11
\n\u2234 The points of intersection are (0, 1), (0, -1).
\nThe parabola meets the X- axis and y = 1 and y = – 1 and the curve is symmetric with respect to X – axis
\n\"TS<\/p>\n

Question 21.
\nFind the area enclosed by the curves y = 3x and y = 6x-x2<\/sup>.
\nSolution:
\nGiven curves are y3<\/sup>x and y=6x – x2
\n<\/sup>Solving 6x – x2<\/sup> = 3x = 3x – x2<\/sup> = 0
\n= x(3- x)=0 =x=0 or x=3
\nTaking f(x) = 3x and g(x) = 6x – x2<\/sup>
\nthen g(x) \u2265 1(x) in [0, 3] and area enclosed between the line y = 3x and the parabola y = 6x-x2<\/sup> is
\n\"TS<\/p>\n

Long Answer Type Questions<\/span><\/p>\n

Question 1.
\nShow that \\(\\int_0^{\\frac{\\pi}{2}} \\frac{x}{\\sin x+\\cos x}\\) dx =\\(\\frac{\\pi}{2 \\sqrt{2}} \\log (\\sqrt{2}+1)\\)
\nSolution:
\n\"TS
\n\"TS
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 2.
\nEvaluate \\(\\int_{\\frac{\\pi}{6}}^{\\frac{\\pi}{3}} \\frac{\\sqrt{\\sin x}}{\\sqrt{\\sin x}+\\sqrt{\\cos x}}\\) dx
\nSolution:
\n\"TS
\n\"TS<\/p>\n

Question 3.
\nEvaluate \\(\\int_{-a}^a\\left(x^2+\\sqrt{a^2-x^2}\\right) dx\\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 4.
\nEvaluate \\(\\int_0^\\pi \\frac{x \\sin x}{1+\\sin x}\\) dx
\nSolution:
\n\"TS
\n\"TS
\n\"TS<\/p>\n

Question 5.
\nFind \\(\\int_0^\\pi \\mathbf{x}\\) sin7<\/sup> x cos 6<\/sup> x dx.
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 6.
\nFind the area enclosed between y=x2<\/sup>-5x and y=4-2x.
\nSolution:
\nThe graphs of curves are shown below.
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 7.
\nFind the area bounded between the curves y = x2<\/sup>, y = \\(\\sqrt{\\mathbf{x}} \\)
\nSolution:
\n\"TS
\n\"TS<\/p>\n

\"TS<\/p>\n

Question 8.
\nFind the area bounded between the curves y2<\/sup>=4ax, x2<\/sup>= 4by(a>0,b>0).
\nSolution:
\n\"TS
\n\"TS<\/p>\n","protected":false},"excerpt":{"rendered":"

Students must practice these\u00a0TS Inter 2nd Year Maths 2B Important Questions Chapter 7 Definite Integrals to help strengthen their preparations for exams. TS Inter 2nd Year Maths 2B Definite Integrals Important Questions Very Short Answer Type Questions Question 1. Evaluate dx Solution: Question 2. Evaluate sinx dx Solution: Question 3. Evaluate Solution: Question 4. Evaluate … Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/14664"}],"collection":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/comments?post=14664"}],"version-history":[{"count":10,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/14664\/revisions"}],"predecessor-version":[{"id":14789,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/14664\/revisions\/14789"}],"wp:attachment":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/media?parent=14664"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/categories?post=14664"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/tags?post=14664"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}