Class 12

Math

Calculus

Area

Area of the region bounded by the curve $y=e_{x_{2}}$ and the line y=e is

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Let $O(0,0),A(2,0),andB(1,3 1 $ be the vertices of a triangle. Let R be the region consisting of all theose points P inside $ΔOAB$ which satisfy $d(P,OA)≤min[d(P,AB)],$ where d denotes the distance from the point to the corresponding line. Sketch the region R and find its area.

Area bounded by the min. ${∣x∣,∣y∣}=1$ and the max. ${∣x∣,∣y∣}=2$ is

The area of the region ${(x,y):x_{2}+y_{2}≤5,∣∣x∣−∣y∣∣≥1$ is

Find the area bounded by $y=∣∣ sinx−21 ∣∣ andy=1forx∈[0,π]$

$Two curvesC_{1}≡[f(y)]_{2/3}+[f(x)]_{1/3}=0andC_{2}≡[f(y)]_{2/3}+[f(x)]_{2/3}=12,satisfying the relation(x−y)f(x+y)−(x+y)f(x−y)=4xy(x_{2}−y_{2})$ The area bounded by the curve $C_{2}and∣x∣+∣y∣=12 $ is

The area bounded by the curve $a_{2}y=x_{2}(x+a)$ and the x-axis is

Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., $x=x(t),y=(t),$ then the area of the figure is evaluated by one of the three formulas : $S=−α∫ β y(t)x_{′}(t)dt,$ $S=α∫ β x(t)y_{′}(t)dt,$ $S=21 α∫ β (xy_{′}−yx_{′})dt,$ Where $αandβ$ are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t. The area of the region bounded by an are of the cycloid $x=a(t−sint),y=a(1−cost)$ and the x-axis is

The area of the figure bounded by the parabola $(y−2)_{2}=x−1,$ the tangent to it at the point with the ordinate x=3, and the x-axis is