What is area y=7x,y= 7/4 x,y=78-x^2 ?

The answer to area y=7x,y= 7/4 x,y=78-x^2 is 59619/128

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

area y=7x,y= 7/4 x,y=78-x^2

area

y = 7 x, y = \frac{7}{4} x, y = 78 - x^{2}

\frac{59619}{128}

Decimal

465.77343 \dots

Solution steps

To find the intersection points solve

f_{i} (x) = f_{j} (x)

7 x = \frac{7}{4} x : x = 0

7 x = 78 - x^{2} : x = - 13, x = 6

\frac{7}{4} x = 78 - x^{2} : x = - \frac{39}{4}, x = 8

= \int_{- 13}^{- \frac{39}{4}} ∣ f_{1} (x) - f_{3} (x) ∣ d x + \int_{- \frac{39}{4}}^{0} ∣ f_{2} (x) - f_{1} (x) ∣ d x + \int_{0}^{6} ∣ f_{1} (x) - f_{2} (x) ∣ d x + \int_{6}^{8} ∣ f_{2} (x) - f_{3} (x) ∣ d x

= \int_{- 13}^{- \frac{39}{4}} 7 x - (78 - x^{2}) d x + \int_{- \frac{39}{4}}^{0} \frac{7}{4} x - 7 x d x + \int_{0}^{6} 7 x - \frac{7}{4} x d x + \int_{6}^{8} \frac{7}{4} x - (78 - x^{2}) d x

\int_{- 13}^{- \frac{39}{4}} 7 x - (78 - x^{2}) d x = \frac{17069}{192}

\int_{- \frac{39}{4}}^{0} \frac{7}{4} x - 7 x d x = \frac{31941}{128}

\int_{0}^{6} 7 x - \frac{7}{4} x d x = \frac{189}{2}

\int_{6}^{8} \frac{7}{4} x - (78 - x^{2}) d x = \frac{197}{6}

= \frac{17069}{192} + \frac{31941}{128} + \frac{189}{2} + \frac{197}{6}

Simplify

= \frac{59619}{128}

\int - \frac{1}{50} d x

\int_{0}^{1} \frac{5 x ^{4}}{1 + x ^{5}} d x

\int (6 x - 2)^{- 4} d x

\int \frac{2 r}{( 1 - r ) ^{7}} d r

\int - \frac{2}{1 - 9 x ^{2}} d x

What is area y=7x,y= 7/4 x,y=78-x^2 ?
The answer to area y=7x,y= 7/4 x,y=78-x^2 is 59619/128