What is the Area Under a Curve? 🤔
The **area under a curve** represents the accumulated value of a quantity over an interval. In calculus, this is one of the main applications of the **definite integral**.
If a function $f(x)$ is positive on an interval $[a, b]$, the definite integral $\int_a^b f(x) \,dx$ gives the exact area between the curve, the x-axis, and the vertical lines at $x=a$ and $x=b$.
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Video: Approximating the Area 🎬
Before integrals, mathematicians used rectangles to approximate the area. Watch this video to see how this works!
Practice Problems
Use the Fundamental Theorem of Calculus to find the area under the curve.
1) Find the area under $f(x) = 2x$ from $x=0$ to $x=4$.
2) Find the area under $f(x)=x^2$ from $x=0$ to $x=3$.
3) Find the area under $f(x)=x^3$ from $x=1$ to $x=2$.
4) Find the area under $f(x)=5$ from $x=0$ to $x=6$.
⚡ Area Under a Curve Quiz
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