Given the function $$ f(x)=12 x^{5}-4 x^{3}+2 x $$ for $$ x $$ between -1 and 2, find: The net area $$ =[?] $$ The total area $$ =[ $$

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We are given $$ f(x)=12x^5-4x^3+2x $$ for $$ x $$ in the interval $$[-1,2]$$. ### Step 1. Find the antiderivative We integrate term‐by‐term: $$ \int 12x^5\,dx = 2x^6, $$ $$ \int (-4x^3)\,dx = -x^4, $$ $$ \int 2x\,dx = x^2. $$ Thus, an antiderivative $$ F(x) $$ is $$ F(x)=2x^6 - x^4 + x^2. $$ ### Step 2. Compute the Net Area The net area is given by $$ \int_{-1}^{2} f(x)\,dx = F(2)-F(-1). $$ First, compute $$ F(2) $$: $$ F(2)=2(2)^6 - (2)^4 + (2)^2 = 2\cdot64 - 16 + 4 = 128 - 16 + 4 = 116. $$ Next, compute $$ F(-1) $$: $$ F(-1)=2(-1)^6 - (-1)^4 + (-1)^2 =2\cdot1 - 1+1 = 2-1+1=2. $$ Thus, the net area is $$ F(2)-F(-1)=116-2=114. $$ ### Step 3. Determine Where the Function Changes Sign We need the total area, which is the sum of the absolute areas where $$ f(x) $$ may be negative. Notice that $$ f(x)=12x^5-4x^3+2x = x\left(12x^4-4x^2+2\right). $$ The factor $$12x^4-4x^2+2$$ is always positive for all real $$ x $$ (its discriminant is negative when viewed as a quadratic in $$ x^2 $$). Hence, the sign of $$ f(x) $$ is determined by the factor $$ x $$: - For $$ x<0 $$, $$ f(x)<0 $$. - For $$ x>0 $$, $$ f(x)>0 $$. Thus, the only sign change in the interval $$[-1,2]$$ is at $$ x=0 $$. ### Step 4. Compute the Total Area The total area is $$ \text{Total Area}=\int_{-1}^{0} \left|f(x)\right|\,dx+\int_{0}^{2} \left|f(x)\right|\,dx. $$ Since $$ f(x) $$ is negative on $$[-1,0]$$ and positive on $$[0,2]$$: $$ \int_{-1}^{0} \left|f(x)\right|\,dx = -\int_{-1}^{0} f(x)\,dx, $$ $$ \int_{0}^{2} \left|f(x)\right|\,dx = \int_{0}^{2} f(x)\,dx. $$ #### Compute $$\int_{-1}^{0} f(x)\,dx$$ $$ \int_{-1}^{0} f(x)\,dx=F(0)-F(-1)=0-2=-2. $$ Taking the absolute value gives $$ 2 $$. #### Compute $$\int_{0}^{2} f(x)\,dx$$ $$ \int_{0}^{2} f(x)\,dx=F(2)-F(0)=116-0=116. $$ Therefore, the total area is $$ 2+116=118. $$ ### Final Answers Net area: $$ 114 $$ Total area: $$ 118 $$ Net area: 114 Total area: 118

Given the function

for between -1 and 2, find:
The net area
The total area

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1. Find the antiderivative

Step 2. Compute the Net Area

Step 3. Determine Where the Function Changes Sign

Step 4. Compute the Total Area

Compute

Compute

Final Answers

Mind Expander

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Latest Calculus Questions

Given the function for between -1 and 2, find: The net area The total area

Solución de inteligencia artificial de Upstudy

Responder

Solución

Step 1. Find the antiderivative

Step 2. Compute the Net Area

Step 3. Determine Where the Function Changes Sign

Step 4. Compute the Total Area

Compute

Compute

Final Answers

Mind Expander

preguntas relacionadas

Latest Calculus Questions

Given the function

for between -1 and 2, find:
The net area
The total area