Concept explainers
Area under a curve Suppose the function y = h(x) is nonnegative and continuous on [α, β], which implies that the area bounded by the graph of h and x-axis on [α, β] equals
105. Find the area of the region bounded by the asteroid x = cos3 t, y = sin3 t, for 0 ≤ t ≤ 2π (see Example 8, Figure 12.17).
Want to see the full answer?
Check out a sample textbook solutionChapter 12 Solutions
Calculus: Early Transcendentals (3rd Edition)
Additional Math Textbook Solutions
Precalculus
Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book
Precalculus Enhanced with Graphing Utilities (7th Edition)
Calculus & Its Applications (14th Edition)
- Write parametric equations for a cycloid traced by a point P on a circle of radius a as the circle rolls along the x -axis given that P is at a maximum when x=0.arrow_forwardThe asymptotes of the graph of the two given parametric equations are * 1 and t X = y = t +1 O x=0 , y=0 x = 0 only x = -1, y = 0 O x = - 1 only O x = 0, y = 1arrow_forwardDIFFERENTIATION OF PARAMETRIC EQUATIONS d²y 1.) of y = sinInß and x = In(Inß) dx2 и 2.) dx2 of y = costu – sin*u and x = 2cos? 2 3.) of s = and t = k-1 k+1 d?x 4.) of x = (02 – 1)2 and y = 403 d²y 5.) dx2 of x = 1- Int and y t-Int %D 2arrow_forward
- A curve Cis given by the parametric equations z= 2t –t2 and y = (t- 2)* wheret > 0. 1. The curve has a vertical tangent line at the point O(0,-2) O(1,0) O(0, -1) O(0, 4) 2. Is the point (-8, 8) lies on the curve C? O(0,0) O(1,-1) OYes ONOarrow_forwardIt can be shown that the parametric equations x = x1 + (x2 – x1)t, y = y1 + (y2 – yı)t, where 0arrow_forward_1. The Cartesian equation of the curve given in parametric equations x = e' – e-t , y = e' + e¬t is (a) x² – y² = 4. (c) x² – y² = 1. (d) y² – x² = 1. (b) y² – x² = 4. _2. The equation of the tangent line to the curve x = -2t² + 3, y = t³ – 8t at the point where t = -2 is (a) x – 2y +7 = 0. (b) x + 2y – 7 = 0. (c) 2x – y + 7 = 0. (d) None of these. _3. The slope m of the tangent line at any point on the curve r = 2 cos 0 is (b) – cot 20. (a) – tan 20. (c) tan 20. (d) None of these. _4. The graph of the equation r² = 8 cos20 is a (a) limacon (c) circle (b) cardioid. (d) lemniscate. _5. The distance between the points A(-2, –3,1) and B(6,9,-3) is (b) 2/14 units . (a) 4v14 units . (c) 8/14 units . (d) 4/7 units.arrow_forwardFind the values of h, k, and a that make the circle (x - h)2 + (y - k)2 = a2 tangent to the parabola y = x2 + 1 at the point (1, 2) and that also make the second derivatives d2y/dx2 have the same value on both curves there. Circles like this one that are tangent to a curve and have the same second derivative as the curve at the point of tangency are called osculating circles.arrow_forwardThe parametric equation of the tangent line to the curve C defined by the following parametric equation x =t y =ť² + 2 „where t e (-00, 0), at t = 1 is Select one: x = t, y = 2t, t E (-00, 00) x =t +1, y = 2t, t E (-∞, 0) x =t+2,y = 2t, t E (-∞0, ∞) x = t, y = 2t + 1, t E (-∞0, 0) O x = t, y = 2t + 2, t E (-∞, a)arrow_forwardEvaluate S I (* + 7)i + (2») 5).dF. (x + 7) where Lis the quarter-circle with parametric form z = cos (t) and y = sin (t), where 0 tarrow_forwardIn Basic Calculus. Thank youarrow_forwardy=t^2+4t See question in imagearrow_forwardA parametric equation of a line in R3 through the point (zo, yo, zo) and parallel to another line with the following parametric equation x = x₁ + at y = y₁ + bt z = 2₁ + ct is 0 x= = (x₁ - x0) + at y = (y₁ - yo) + bt Z= = (2₁-zo) + ct x = x₁ + at y=yo+bt z = 20 + ct tER x = x₁ + (x₁ -xo)t y = yo + (y₁ - yo)t te R z = 20 + (21 - 2o)t tER x= xo + (yoc - zob)t y=yo+ (zoa - xoc)t te R 2= 20 + (zob - yo a)t H= = (xo-x₁) + at y = (30-3₁) + bt z = (20 - 2₁) + ct tER tERarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage