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SolverIntegralsMedium
Exercise #22
Definite integral — area under a curve
∫20(x2 + 1) dx

Calculate the area under f(x) = x² + 1 on the interval [0, 2].

02f(x) = x² + 1A
  1. Step 1: Find an antiderivative F(x) of f(x) = x² + 1.Solved
    F(x) = x33 + x
  2. Step 2: Calculate F(2), the value of the antiderivative at x = 2.
  3. Step 3: Calculate F(0), the value of the antiderivative at x = 0.
  4. Step 4: Apply the fundamental theorem: area = F(2) − F(0).
subjects
6
chapters
60
worked examples & solved exercises
340+
guided exercises
98
languages
4

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SolverLimitsMedium
Exercise #15
Notable limit with sin(ax)/(bx)
limx→0sin(4x)2x

Calculate the limit using the fundamental notable limit.

y = 20
  1. Step 1: Multiply and divide by 4 so that sin(4x)/(4x) appears: which coefficient is left in front?
  2. Step 2: With t = 4x, t → 0 as well: what is the limit of sin(4x)/(4x)?
  3. Step 3: Multiply the coefficient by the notable limit: what is the limit?

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Simone Rossato

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Content written and checked with care

Uningenia is written and checked by hand by Simone Rossato, an engineer: the goal is to explain the why behind every formula, not just the how. The exercises mirror typical university exam problems and are solved with every intermediate step, so you can follow the reasoning, not just the final answer.

Simone RossatoEngineer · founder of Uningenia
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Studying engineering with Uningenia

Uningenia is the free study platform for engineering and science students: complete theory, interactive charts and hundreds of step-by-step solved exercises. It covers Physics 1 (mechanics, fluids, thermodynamics, waves), Physics 2 (electromagnetism and optics), Calculus I and II, General Chemistry and Organic Chemistry.

Every chapter combines clear explanations, ready-to-use formulas and diagrams that adapt to the light or dark theme. Download the formula sheets as PDF, try the exercise solver and build concept maps. All material is available in Italian, English, Spanish and French, with no sign-up and completely free.

The material is aimed at first- and second-year engineering, physics, mathematics and chemistry students, but is also useful for high-school revision. Everything is free, with no sign-up: the project is supported only by small affiliate links to recommended books. Found a mistake or want to suggest a topic? Write to rossimo.developer@gmail.com.

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