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The equation of curve is y = sin x. Its rough sketch from x = 0 to x - 2

straight pi

is shown is the figure.

Required area =

integral subscript 0 superscript straight pi space sinx space dx space plus space integral subscript straight pi superscript 2 straight pi end superscript left parenthesis negative sin space straight x right parenthesis space dx

open curly brackets because space sin space straight x space greater or equal than 0 space for space straight x space element of space left square bracket 0 comma space straight pi right square bracket space and space sin space straight x space less or equal than space 0 space for space straight x space element of space left square bracket straight pi comma space 2 straight pi close curly brackets

equals space left square bracket negative cos space straight x right square bracket subscript 0 superscript straight pi plus open square brackets cos space straight x close square brackets subscript straight pi superscript 2 straight pi end superscript space equals space left parenthesis negative cos space straight pi space plus space cos space 0 right parenthesis space plus space left parenthesis cos space 2 straight pi space minus space cos space straight pi right parenthesis equals space left parenthesis 1 plus 1 right parenthesis plus left parenthesis 1 plus 1 right parenthesis space equals space 4 space sq. space units.

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42.

Using integration, find the area of the triangular region whose sides have the equations y = 2 x + 1, y = 3 x + 1 and x = 4.

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43. Using the method of integration find the area of the region bounded by lines:
2 x + y = 4, 3 x - 2 y = 6 and x - 3 y + 5 = 0.

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44.

Using integration, find the area of the region bounded by (2, 5), (4, 7) and (6, 2).

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45. Using the method of integration, find the area of the triangle ABC, co-ordinates of whose vertices are A (2, 0), B (4, 5), C (6, 3).

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46.

Using integration, find the area of the region bounded by the triangle whose vertices are (-1, 1), (0, 5) and (3, 2).

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47. Using integration, find the area of the triangle ABC whose vertices have coordinates A (3, 0), B(4, 6) and C (6, 2).

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48. Using integration, find the area of the triangle ABC whose vertices are A (3, 0) B (4, 5) and C (5, 1).

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49.

Using integration, find the area of the region bounded by the triangle whose vertices are (1, 0), (2, 2) and (3, 1).

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50.

Using integration find the area of region bounded by the triangle whose vertices are (-1, 0), (1, 3) and (3, 2).

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